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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf,len=1059
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='02233v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='OA]  5 Jan 2023  THE STABLE EXOTIC CUNTZ ALGEBRAS ARE HIGHER-RANK  GRAPH ALGEBRAS  JEFFREY L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' BOERSEMA AND SARAH L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' BROWNE AND ELIZABETH GILLASPY  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' For each odd integer n ≥ 3, we construct a rank-3 graph Λn with involution γn  whose real C∗-algebra C∗  R (Λn, γn) is stably isomorphic to the exotic Cuntz algebra E  R  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' This  construction is optimal, as we prove that a rank-2 graph with involution (Λ, γ) can never  satisfy C∗  R (Λ, γ) ∼ME E  R  n, and the first author reached the same conclusion for rank-1 graphs  (directed graphs) in [Boe17, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Our construction relies on a rank-1 graph with  involution (Λ, γ) whose real C∗-algebra C∗  R (Λ, γ) is stably isomorphic to the suspension SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  In the Appendix, we show that the i-fold suspension SiR is stably isomorphic to a graph  algebra iff −2 ≤ i ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Introduction  For every odd integer n ≥ 3, the (complex) Cuntz algebra On has two real forms: the real  Cuntz algebra O  R  n and the exotic Cuntz algebra En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' While the existence of En follows from the  classification of simple purely infinite real C∗-algebras [Boe06, BRS11], the non-constructive  nature of the existence portion of this classification theorem [Boe06, Theorem 1] means that  we know very little about En beyond its K-theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' In particular, until now there has been  no construction or representation of En in terms of familiar C∗-algebraic objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  In this paper, we give an explicit realization of the stabilized exotic Cuntz algebras KR ⊗R  En as higher-rank graph algebras associated to rank-3 graphs with involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Given the  extensive literature on the properties of higher-rank graph C∗-algebras, we anticipate that  this concrete description will facilitate an improved understanding of these elusive algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  Higher-rank graphs, or k-graphs, are a k-dimensional generalization of directed graphs  which were introduced by Kumjian and Pask in [KP00].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Many of the properties of (complex)  directed graph C∗-algebras, such as their K-theory [RS04] and their ideal structure [BHRS02,  HS04], are visible from the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  While the structure of k-graph C∗-algebras is more  intricate than that of graph C∗-algebras, k-graph C∗-algebras also encompass a broader  range of examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Indeed [RSS15], every complex UCT Kirchberg algebra is a direct limit  of 2-graph C∗-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' The real C∗-algebra C∗  R(Λ, γ) of a higher-rank graph with involution  (Λ, γ) was recently introduced by the first and third authors in [BG22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' In that paper, the  authors also generalized the work of [Eva08] and [Boe17] to describe a spectral sequence  which converges to the CR K-theory of these real C∗-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  The main result (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='3) of the present paper, that the exotic Cuntz algebra is stably  isomorphic to the C∗-algebra of a 3-graph with involution, is the best possible in terms of  the rank of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' In [Boe17], the first author made an extensive analysis of the K-theory of the  real C∗-algebra C∗  R(Λ, γ) of a rank-1 graph (directed graph) with involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' 1 In particular,  [Boe17, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='3] establishes that the exotic Cuntz algebra cannot be isomorphic or  stably isomorphic to the real C∗-algebra C∗  R(Λ) of a directed graph Λ, or to the real C∗-  algebra C∗  R(Λ, γ) of a graph with involution, since KO7(En) = Z2 but KO7(C∗  R(Λ, γ)) is  always torsion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='1 below uses the K-theory spectral sequence for real higher-  rank graph C∗-algebras ([BG22, Section 3]) to show that En ̸∼ME C∗  R(Λ, γ)) for any rank-2  1This class of C∗-algebras includes the real C∗-algebras C∗  R (Λ) of a directed graph, introduced in [Boe14],  as C∗  R (Λ) ∼= C∗  R (Λ, γtriv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  1  2  JEFFREY L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' BOERSEMA AND SARAH L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' BROWNE AND ELIZABETH GILLASPY  graph with involution (Λ, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  However, we construct in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='3 a family of rank-3  graphs with involution (Λn, γn) such that C∗  R(Λn, γn) ∼= En ⊗R KR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  Our construction combines the directed graphs with involution (En, γn) of [Boe17, Exam-  ple 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='2], which satisfy C∗  R(En, γn) ∼ME S6En, with a directed graph with involution (Λ, γ)  such that (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='1) C∗  R(Λ, γ) is a real Kirchberg algebra which is KK-equivalent  to the suspension algebra SR ∼= C0((0, 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  To be precise, the 3-graph Λn which gives  C∗  R(Λn, γn) ∼= En ⊗R KR is a product graph, Λn = En × Λ × Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  Prompted by the graph with involution of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='1, we consider in Section 5 the  question of which suspensions SiR are KK-equivalent to the real C∗-algebra of a graph with  involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' For −2 ≤ i ≤ 1 we exhibit an example of a graph with involution (Λ, γ) such that  C∗  R(Λ, γ) is KK-equivalent to SiR, and we show in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='2 that SiR ̸∼KK C∗  R(Λ, γ)  if 2 ≤ i ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' (However, we can realize these suspensions as 2-graph or 3-graph algebras, by  taking products of the graphs which do realize suspensions of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=')  Many key questions remain open for further investigation about the class of real C∗-  algebras that can be obtained using higher-rank graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' For example, it is still unknown  whether or not En itself can be realized as a rank-k graph-with-involution algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Similarly,  it remains unknown which real Kirchberg algebras can be realized by higher-rank graphs  with involution (as opposed to inductive limits of such objects);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' we would particularly like  to find a K-theoretic characterization of such algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  Acknowledgments: E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' was partially supported by NSF grant 1800749.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Preliminaries  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Higher-rank graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' [KP00, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='1] A higher-rank graph of rank k, or a k-graph, is  a countable small category Λ equipped with a degree functor d: Λ → Nk such that, if a  morphism λ ∈ Λ satisfies d(λ) = m + n, then there exist unique morphisms µ, ν ∈ Λ such  that λ = µν, d(µ) = m and d(ν) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  Write ei for the standard ith basis vector of Nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  The morphisms of degree ei can be  advantageously viewed as the “edges of color i” in Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' In this perspective, if e is an edge of  color i and f is an edge of color j, their composition ef ∈ Λ satisfies  d(ef) = ei + ej = ej + ei,  so we must be able to rewrite ef = f ′e′ for some morphisms e′, f ′ ∈ Λ with d(f ′) = ej and  d(e′) = ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  Indeed, by [HRSW13, Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='4 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='5], a k-graph can be equivalently thought of  as arising from a directed graph G, with k colors of edges and with a factorization rule on  multicolored paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' That is, given any two colors (“red��� and “blue”) and any two vertices  v, w in G, the factorization rule identifies each red-blue path ef from v to w with an equivalent  blue-red path f ′e′ from v to w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  We would like the quotient of the space G∗ of directed paths in G by the equivalence  relation ∼ generated by the factorization rule to be a k-graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  For this to occur, the  factorization rule must also satisfy certain consistency conditions which ensure that, for each  path in G∗, its equivalence class under ∼ corresponds to a k-dimensional hyper-rectangle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  see [EFG+21, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='3] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' As our work in this paper does not depend on  these consistency conditions, we will not reproduce them here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' That said, we remark that in  THE STABLE EXOTIC CUNTZ ALGEBRAS ARE HIGHER-RANK GRAPH ALGEBRAS  3  a rank-1 graph, the factorization rule is nonexistent, and so a 1-graph is precisely the space  of paths of a directed graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  Let Λ be a k-graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Given n ∈ Nk and objects v, w ∈ Λ, we write  (1)  Λn = {λ ∈ Λ : d(λ) = n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  By the factorization rule, for every λ ∈ Λ, there are unique v, w ∈ Λ0 with vλ = λw =  vλw = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' That is, we can identify Λ0 with the objects of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' If λ = vλw, we write v = r(λ)  and w = s(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Thus, expanding on Equation (1), we have  vΛn = {λ ∈ Λ : r(λ) = v and d(λ) = n}  Λnw = {λ ∈ Λ : s(λ) = w and d(λ) = n},  (2)  as well as the obvious variations such as vΛnw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  A k-graph Λ has k adjacency matrices Mi ∈ MΛ0(N), which are given by  (3)  Mi(v, w) = #vΛeiw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  In the graphical picture, λ ∈ Λ(n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=',nk) means that λ represents the ∼-equivalence class of  a path with ni edges of color i, for each 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' That is, Λ0 consists of the length-0 paths,  ie, the vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Then Mi(v, w) is the number of edges of color i from vertex w to vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  If Λ1 is a k1-graph and Λ2 is a k2-graph, then [KP00, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='8] their (Cartesian)  product Λ1 ×Λ2 is a (k1 +k2)-graph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' the degree functor is given by d(λ1, λ2) = (d(λ1), d(λ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  We have (Λ1 × Λ2)0 = Λ0  1 × Λ0  2 and s(λ1 × λ2) = (s(λ1), s(λ2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  In this paper we will focus on k-graphs which are row-finite and source-free (or have no  sources).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' We say a k-graph Λ is row-finite if |vΛn| < ∞ for all n ∈ Nk and v ∈ Λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' The  k-graph has no sources if vΛn ̸= ∅ for all v, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' It is straightforward to check that if Λ1, Λ2  are row-finite and source-free, then so is Λ1 × Λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  For a row-finite source-free k-graph Λ, its (complex) C∗-algebra is the universal C∗-algebra  generated by a Cuntz–Krieger Λ-family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' [KP00, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='5] Given a row-finite source-free k-graph Λ, a Cuntz–Krieger  Λ-family is a collection {tλ}λ∈Λ of partial isometries in a C∗-algebra A which satisfy the fol-  lowing conditions:  (CK1) For each v ∈ Λ0, tv is a projection, and tvtw = δv,wtv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  (CK2) For each λ ∈ Λ, t∗  λtλ = ts(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  (CK3) For each λ, µ ∈ Λ, tλtµ = tλµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  (CK4) For each v ∈ Λ0 and each n ∈ Nk, tv =  �  λ∈vΛn  tλt∗  λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  We define C∗(Λ) to be the universal (complex) C∗-algebra generated by a Cuntz–Krieger  family, in the sense that for any Cuntz–Krieger Λ-family {tλ}λ∈Λ, there is a surjective ∗-  homomorphism C∗(Λ) → C∗({tλ}λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  We write {sλ}λ∈Λ for the generators of C∗(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' By using the Cuntz–Krieger relations,  one can compute that C∗(Λ) = span{sλs∗  µ : s(λ) = s(µ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='5(iv) of [KP00] also  establishes that C∗(Λ1 × Λ2) ∼= C∗(Λ1) ⊗ C∗(Λ2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' the isomorphism takes s(λ1,λ2) to sλ1 ⊗ sλ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  We will use one more ingredient – an involution – to construct the real C∗-algebras asso-  ciated to higher-rank graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' An involution γ on a k-graph Λ is a degree-preserving functor γ : Λ → Λ  which satisfies γ ◦ γ = id Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  4  JEFFREY L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' BOERSEMA AND SARAH L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' BROWNE AND ELIZABETH GILLASPY  As established in [BG22, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='4], the real C∗-algebra associated to a k-graph Λ and  an involution γ : Λ → Λ is  (4)  C∗  R(Λ, γ) = spanR{zsλs∗  µ + zsγ(λ)s∗  γ(µ) | z ∈ C, λ, µ ∈ Λ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  Equivalently, we have C∗  R(Λ, γ) = {a ∈ C∗(Λ) | �γ(a) = a∗}, where �γ is the antimultiplicative  C∗-involution uniquely determined by �γ(sλ) = s∗  γ(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' For any involution γ on Λ, we have  C∗(Λ) ∼= C ⊗R C∗  R(Λ, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' CRT K-theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' In our work, we will use the full united K-theory K  CRT(A) (introduced  in [Boe02]) as well as the abbreviated variation K  CR(A) which contains just the real and  complex parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='2 of [BRS11] shows that the category of real purely infinite  simple C∗-algebras, whose complexifications are simple and in the UCT class, is classified up  to isomorphism by either of these invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' We tend to use K  CR(A) since it is simpler and  usually sufficient, but we will also need to use K  CRT(A) on occasion since that is the context  in which we have the K¨unneth formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Specifically, recall that for a real C∗-algebra A,  K  CR(A) = {KO∗(A), KU∗(A)}  K  CRT(A) = {KO∗(A), KU∗(A), KT∗(A)}  where KO∗(A) is the standard 8-periodic real K-theory for a real C∗-algebra and KU∗(A) =  K∗(C ⊗C A) is the 2-periodic K-theory of the complexification of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Meanwhile KT∗(A) is  the 4-periodic self-conjugate K-theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' These invariants also include the additional CR and  CRT -module structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' In particular for K  CR(A) there are natural transformations  ri : KUi(A) → KOi(A)  induced by the standard inclusion C → M2(R)  ci : KOi(A) → KUi(A)  induced by the standard inclusion R → C  ψi : KUi(A) → KUi(A)  induced by conjugation C → C  ηi : KOi(A) → KOi+1(A)  induced by multiplication by η ∈ KO1(R) = Z2  ξi : KOi(A) → KOi+1(A)  induced by multiplication by ξ ∈ KO4(R) = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  This additional structure tends to aid in the computations of KO∗(A) because the natural  transformations satisfy the relations  rc = 2  cr = 1 + ψ  2η = 0  rψ = r  ψ2 = id  η3 = 0  ψc = c  ψβU = −βUψ  ξ = rβ2  Uc  and they fit into a long exact sequence  (5)  · ·  rβ−1  U  −−−→ KOi(A)  η−→ KOi+1(A)  c−→ KUi+1(A)  rβ−1  U  −−−→ KOi−1(A)  η−→ · · ·  These two invariants K  CR(A) and K  CRT(A) contain the same information by results of  [Hew96] (summarized, for example, in [BRS11, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' K-theory for higher-rank graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' For the real C∗-algebra C∗  R(Λ, γ) of a higher-  rank graph with involution, [BG22, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='10] establishes the existence of a spectral  sequence {Er, dr} of CR-modules that converges to K  CR(C∗  R(Λ, γ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' The complex part of this  spectral sequence (Er  p,q)  U coincides with the Evans spectral sequence [Eva08] and converges  to KU∗(C∗  R(Λ, γ)) = K∗(C∗(Λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' The real part of this spectral sequence (Er  p,q)  O converges to  KO∗(C∗  R(Λ, γ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  THE STABLE EXOTIC CUNTZ ALGEBRAS ARE HIGHER-RANK GRAPH ALGEBRAS  5  The E2 page of the spectral sequence arises from the homology of a certain chain complex C  based on the combinatorial information of Λ and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' We will use the spectral sequence only in  the rank-1 and rank-2 cases, where these chain complexes have the following straightforward  descriptions which we recall from [BG22, Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='14 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' First let Λ0  f be the set  of vertices fixed by the involution γ and let Λ0  g ⊔ Λ0  h be any partition of Λ0\\Λ0  f satisfying  γ(Λ0  g) = Λ0  h and γ(Λ0  h) = Λ0  g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Then let A = K  CR(R)Λ0  f ⊕ K  CR(C)Λ0  g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  For a rank-1 graph with involution the chain complex C is given by  0 → A  ∂1  −→ A → 0,  where ∂1 = ρ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' For a rank-2 graph with involution the chain complex C is given by  0 → A  ∂2  −→ A2 ∂1  −→ A → 0,  where ∂1 =  �  ρ1  ρ2�  and ∂2 =  �  −ρ2  ρ1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Here the maps ρi are determined by the adjacency  structure of Λ as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' For 1 ≤ i ≤ k, the complex part (ρi)  U  0 : ZΛ0 → ZΛ0 is represented  by the matrix Bi = I − Mt  i , where Mi is the adjacency matrix of the graph Λ for the edges  of degree ei, and (ρi)  U  1 = 0 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' The real parts of this map (ρi)  O  j , for 0 ≤ j ≤ 7, can  similarly be determined for each i by some variations of Bi as shown in Table 3 of [BG22]  and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='4 of [Boe17], which we also reproduce here as Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='complex part  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
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+page_content='\uf8f8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='Z|Λ0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='f | ⊕ Z|Λ0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='g| ⊕ Z|Λ0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='h → Z|Λ0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='f| ⊕ Z|Λ0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='g| ⊕ Z|Λ0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='h  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
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+page_content='real part  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
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+page_content='g|  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
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+page_content='f| ⊕ Z|Λ0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
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+page_content='Z|Λ0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='g| → Z|Λ0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='g|  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
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+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Chain complex maps for real K-theory  For reference, the groups of K  CR(R) and K  CR(C) are shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' The natural trans-  formations η, c, r, and ψ that are part of the structure of united K-theory are uniquely  determined from these groups and the long exact sequence (5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' they are also shown in Ta-  bles 1 and 2 of [BG22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' In particular we note that for KO∗(R), the map ηi is non-trivial  exactly for i = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  6  JEFFREY L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' BOERSEMA AND SARAH L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' BROWNE AND ELIZABETH GILLASPY  0  1  2  3  4  5  6  7  KO∗(R)  Z  Z2  Z2  0  Z  0  0  0  KU∗(R)  Z  0  Z  0  Z  0  Z  0  0  1  2  3  4  5  6  7  KO∗(C)  Z  0  Z  0  Z  0  Z  0  KU∗(C)  Z2  0  Z2  0  Z2  0  Z2  0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Non-Existence of a Rank-2 Graph with Involution  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Let n be an odd integer, n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' There does not exist a (row-finite, source-  free) rank-2 graph with involution (Λ, γ) such that K  CR(C∗  R(Λ, γ)) ∼= K  CR(En).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Suppose that (Λ, γ) is a row-finite, source-free rank-2 graph with involution and that  K  CR(C∗  R(Λ, γ)) ∼= K  CR(En).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' For reference we reproduce the groups of K  CR(En) here:  0  1  2  3  4  5  6  7  KO∗(En)  Z2(n−1)  Z2  Z2  0  Z(n−1)/2  0  Z2  Z2  KU∗(En)  Zn−1  0  Zn−1  0  Zn−1  0  Zn−1  0  Note that since KU7(En) = 0 and since im η6 = ker c7 from the long exact sequence (5) relat-  ing KO∗(A) and KU∗(A), we see immediately that η6: Z2 → Z2 must be an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  Now, we consider only the real part of the spectral sequence from [BG22, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  This is a spectral sequence converging to KO∗(C∗  R(Λ, γ)), where the E2-page consists of the  homology of the chain complex  (6)  0 → A  ∂2  −−→ A2  ∂1  −−→ A → 0  where A ∼= KO∗(R)Λ0  f ⊕ KO∗(C)Λ0  g as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  In particular, the spectral sequence has three non-zero columns (for 0 ≤ p ≤ 2) and  is periodic in q (with period 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Furthermore, a quick examination of the structure of A  (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Figure 1) reveals that Ai = 0 for i = 3, 5, 7 and also that Ai is free for i = 0, 4, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  Consequently, E2  p,q = 0 for q = 3, 5, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Moreover, since E2  2,q = ker ∂2 is a subgroup of Ai, we  must have E2  2,q free for q = 0, 4, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Indeed, E∞  2,q = ker d2  2,q is a subgroup of E2  2,q, so E∞  2,q must  also be free for q = 0, 4, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' On the other hand, KOi(C∗  R(Λ, γ)) is finite in all degrees, so it  must be that E∞  2,q = 0 for q = 0, 4, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  THE STABLE EXOTIC CUNTZ ALGEBRAS ARE HIGHER-RANK GRAPH ALGEBRAS  7  From what we have said so far, the E∞  p,q page of the spectral sequence is as follows:  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  7  0  0  0  6  ∗  ∗  0  5  0  0  0  4  ∗  ∗  0  3  0  0  0  2  ∗  ∗  ∗  1  ∗  ∗  ∗  0  ∗  ∗  0  q/p  0  1  2  This E∞ page identifies a filtration of KO∗(C∗  R(Λ, γ)), in which the subquotients of KOj(C∗  R(Λ, γ))  appear along the diagonal p + q = j of the E∞ page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  By hypothesis we have KO0(C∗  R(Λ, γ)) = Z2(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' However, the only non-zero group along  the diagonal p + q = 0 is E∞  0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Thus E∞  0,0 = Z2(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Similarly, since KO7(C∗  R(Λ, γ)) = Z2  and KO6(C∗  R(Λ, γ)) = Z2, we must have E∞  1,6 = Z2 = E∞  0,6:  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  7  0  0  0  6  Z2  Z2  0  5  0  0  0  4  ∗  ∗  0  3  0  0  0  2  ∗  ∗  ∗  1  ∗  ∗  ∗  0  Z2(n−1)  ∗  0  q/p  0  1  2  Now, we consider the natural transformation η: A → A of degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Because A is a  direct sum of copies of K  CR(R) and K  CR(C), which data already includes the map η, the  natural transformation η : KO∗(C∗  R(Λ, γ)) → KO∗+1(C∗  R(Λ, γ)) exists at the level of the  chain complex (6), passes to a map on the E2-page, then to a map on the E∞-page, and  finally converges to the map η: KOi(C∗  R(Λ, γ)) → KOi+1(C∗  R(Λ, γ)) described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  This means that the map η on KO∗(C∗  R(Λ, γ)) respects the filtration of KOi(C∗  R(Λ, γ))  associated with the spectral sequence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' and the resulting maps on the subquotients are the  same as that on the E∞ page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  In particular, the diagonals of the E∞-page that yield  KO6(C∗  R(Λ, γ)) and KO7(C∗  R(Λ, γ)) give the commutative diagram below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' The vertical η  maps on the left and right come from A as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  0  � E∞  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='6  �  η  �  KO6(C∗  R(Λ, γ))  �  η  �  E∞  1,5  �  η  �  0  0  � E∞  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='7  � KO7(C∗  R(Λ, γ))  � E∞  1,6  � 0  ⇐⇒  0  � Z2  �  η  �  Z2  �  η  �  0  �  η  �  0  0  � 0  � Z2  � Z2  � 0  8  JEFFREY L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' BOERSEMA AND SARAH L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' BROWNE AND ELIZABETH GILLASPY  Since the nonzero horizontal maps must be isomorphisms, the commutative diagram forces  the vertical map η6: KO6(C∗  R(Λ, γ)) → KO7(C∗  R(Λ, γ)) in the center of the diagram to be  zero, which contradicts the known value of η in KO∗(E  R  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Existence of a Rank-3 Graph with Involution  In this section, we will construct a 3-graph Λ with involution γ, by taking products of  1-graphs with involution, such that C∗  R(Λ, γ) is stably isomorphic to E  R  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' In what follows, we  use the following convention for suspension of graded modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' If H = {Hi} is a Z-graded  group, then ΣH is a Z-graded group with (ΣH)i = Hi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Similarly, Σ−1H is a Z-graded  group with (Σ−1H)i = Hi−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' This convention is consistent with K-theory and suspensions  of C∗-algebras: K∗(SnA) = ΣnK∗(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  For the proof of the following proposition, we will need a few more preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' For a  graph Λ, a subset X ⊆ Λ0 is hereditary if whenever v ∈ X and vΛw ̸= ∅, then w ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' A cycle  in Λ is a path e1e2 · · ·en with r(e1) = s(en) but, for all 1 ≤ i < n, we have s(ei) ̸= r(ei+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  We say that a cycle has an entrance if there exists 1 ≤ j ≤ n and an edge f ̸= ej with  r(f) = r(ej).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' If the only hereditary subsets of Λ0 are ∅ and Λ0, and every cycle in Λ has an  entrance, then [Szy01, Theorem 12] C∗(Λ) is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Furthermore, if every cycle in Λ has an  entrance, then Λ is aperiodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' There exists a 1-graph Λ and involution γ such that K  CR(C∗  R(Λ, γ)) ∼=  ΣK  CR(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Furthermore, C∗  R(Λ, γ) is simple and purely infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Let Λ be the 1-graph below (which extends infinitely in both directions) and let γ be  the non-trivial involution, which fixes the vertices and edges of the infinite branch on the  left and swaps the vertices and edges of the two infinite branches on the right in the obvious  way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' (In fact γ is the only non-trivial involution on Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=') �❅  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
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+page_content='⑦  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='We begin by showing that C∗  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='R(Λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' γ) is simple and purely infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' It is straightforward to  check that Λ has no nontrivial hereditary subsets, and that every cycle has an entrance, so  simplicity of the complex algebra C∗(Λ) follows from [Szy01, Theorem 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Note further that  for every vertex v ∈ Λ0, there is a vertex w with vΛw ̸= ∅ for some vertex w which supports  a loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' As Λ is aperiodic, one easily checks that the conditions of [KP00, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='9]  are satisfied, and so C∗(Λ) is purely infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Consequently, [BRS11, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='9] implies  that the real C∗-algebra C∗  R(Λ, γ) is also simple and purely infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  We now show that KU0(C∗  R(Λ, γ)) = 0 and KU1(C∗  R(Λ, γ)) = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' (This is the same as  calculating K∗(C∗(Λ)) and does not involve the involution γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=') Let M be the adjacency matrix  for Λ (so Mv,w is the number of edges from w to v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Then KU0(C∗  R(Λ, γ)) ∼= coker (I − Mt),  THE STABLE EXOTIC CUNTZ ALGEBRAS ARE HIGHER-RANK GRAPH ALGEBRAS  9  which can be interpreted as saying that KU0(C∗  R(Λ, γ)) is generated by vertex projection  classes [pv], which are subject only to relations of the form  [pv] =  �  w∈Λ0  Mv,w[pw] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  Let v be one of the vertices of Λ that has a loop, and let w ̸= v be the vertex for which there  is an edge from w to v (in each case there is a unique such w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Then the formula above  gives the relation [pv] = [pv] + [pw], which implies that [pw] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' If [pw] = 0 we will say  that w is a zero vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Now if w is a zero vertex and there is only one edge to w, say from  vertex u, then it follows that u is also a zero vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' More generally, if w is a zero vertex  and all the edges to w are known to emanate from zero vertices except possibly one edge  from vertex u, then it follows that u is also a zero vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Using these principles, it is now  straightforward to work through the graph and to find that every vertex is a zero vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  Hence KU0(C∗  R(Λ, γ)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  We know that KU1(C∗  R(Λ, γ)) ∼= ker(I − Mt), which is to say that  (7)  KU1(C∗  R(Λ, γ)) ∼= NΛ :=  �  α: Λ0 → Z | α(v) =  �  w∈Λ0  Mw,v α(w)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  Let v be one of the vertices of Λ that has a loop, and let w ̸= v be the vertex for which  there is an edge from v to w (in each case there is a unique such w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Then we have the  relation α(v) = α(v) + α(w), which implies that α(w) = 0 for any α ∈ NΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' If α(w) = 0 for  all α ∈ NΛ we will say that w is a null vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Now if w is a null vertex and there is only one  edge emanating from w, say to vertex u, then it follows that u is also a null vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' More  generally, if w is a null vertex and all the edges from w are known to point to null vertices  except possibly one edge to vertex u, then u is also a null vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Using these principles,  it is now straightforward to work through the graph and to find that every vertex is a null  vertex, except for the six vertices labelled u, v, w, x, y, z shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='v•  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='�❇  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
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+page_content='⑧  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='Using Equation (7) and the fact that the unlabeled vertices are null vertices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' we see that  any α ∈ NΛ must satisfy the equations  0 = α(u) + α(w)  α(w) = α(v) + α(x)  0 = α(w) + α(x)  α(x) = α(w) + α(y)  0 = α(x) + α(z)  10  JEFFREY L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' BOERSEMA AND SARAH L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' BROWNE AND ELIZABETH GILLASPY  Solving this system over Z, we find that α(u) is a free variable and that  α(v) = −2α(u), α(w) = −α(u), α(x) = α(u), α(y) = 2α(u), and α(z) = −α(u) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  Thus NΛ ∼= Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Hence KU∗(C∗  R(Λ, γ)) = K∗(C∗(Λ)) = (0, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  Turning to the real K-theory, we now prove that KO∗(C∗  R(Λ, γ))) = (Z2, Z2, 0, Z, 0, 0, 0, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  First, we show that the real and complex E2 = E∞ page of the Evans spectral sequence for  C∗  R(Λ, γ) is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  E2  p,q  (8)  real part  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  7  0  0  6  0  Z  5  0  0  4  0  0  3  0  0  2  0  Z  1  Z2  0  0  Z2  0  q/p  0  1  complex part  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  7  0  0  6  0  Z  5  0  0  4  0  Z  3  0  0  2  0  Z  1  0  0  0  0  Z  q/p  0  1  (9)  We have already discussed the complex part of this spectral sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' For the real part we  will only discuss the computations for the rows corresponding to j = −1, 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' As we will see,  this is enough to determine KO∗(C∗  R(Λ, γ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' The other rows can be computed using similar  methods and we include them in the table above for completeness, but we will neither need  nor discuss them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  First, the spectral sequence for a 1-graph with involution always vanishes in row j = −1,  since the chain complex vanishes in that degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' To compute row j = 1 of the spectral  sequence, we refer to [BG22, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='14] and Table 1 above, which indicates that E2  0,1  and E2  1,1 are the cokernel and kernel of the map  (∂1)1 = I − Mt  11 : Z  Λ0  f  2  → Z  Λ0  f  2  where Λ0  f is the set of fixed vertices of (Λ, γ) and M11 is the restriction of the incidence  matrix to those vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' So it suffices to consider the graph consisting of the fixed points of  Λ, shown here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' �  �❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄ �  �❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄  x  �  �❆  ❆  ❆  ❆  ❆  ❆  ❆  ❆  y  �  �❈  ❈  ❈  ❈  ❈  ❈  ❈  ❈  � �  �⑧  ⑧  ⑧  ⑧  ⑧  ⑧  ⑧  ⑧ �  �⑧  ⑧  ⑧  ⑧  ⑧  ⑧  ⑧  ⑧ �  �⑥  ⑥  ⑥  ⑥  ⑥  ⑥  ⑥  ⑥ �  �⑥  ⑥  ⑥  ⑥  ⑥  ⑥  ⑥  z  �  Using this graph, and the same sort of analysis that we did in the complex case, we find that  coker (I − Mt  11) = Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' More precisely, working modulo 2 we find that [pv] = 0 for all vertices  in Λ0  f except those labeled x, y and z in the graph above and that [px] = [py] = [pz] ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' We  also find easily that ker(I − Mt  11) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  Now, for j = 0, we need to find the cokernel and kernel of the map (∂1)0 which we will  do using Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' First recall the partition Λ0 = Λ0  f ⊔ Λ0  g ⊔ Λ0  h where Λ0  f is the set of fixed  THE STABLE EXOTIC CUNTZ ALGEBRAS ARE HIGHER-RANK GRAPH ALGEBRAS  11  vertices (the branch on the left of Λ), Λ0  g is the set of vertices of the “upper right” branch of  Λ, and Λ0  h is the set of vertices of the “lower right” branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' With this structure on Λ, the  (infinite) matrix B = I − Mt can be written in block form as  B = I − Mt =  \uf8eb  \uf8ed  B11  B12  B12  B21  B22  B23  B21  B23  B33  \uf8f6  \uf8f8  where, for example, B12 keeps track of edges from vertices in Λ0  f to vertices in Λ0  g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Using  Table 1, we see that  (∂1)0 : ZΛ0  f ⊕ ZΛ0  g → ZΛ0  f ⊕ ZΛ0  g  is given by  (∂1)0 =  �  B11  2B12  B21  B22 + B23  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  We will use a new graph Λ′ to analyze this map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' The graph Λ′, shown below, is obtained  from Λ by keeping the vertices from Λ0  f and Λ0  g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' For each edge in Λ from a vertex in Λ0  g to  a vertex in Λ0  f, we create a corresponding edge in Λ′ and for each edge in Λ from a vertex  in Λ0  f to a vertex in Λ0  g we create 2 corresponding edges in Λ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Also, for each edge from a  vertex v = γ(u) ∈ Λ0  h to a vertex w ∈ Λ0  g we obtain an edge in Λ′ from u to w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  x  �  �❆  ❆  ❆  ❆  ❆  ❆  ❆  ❆  �  ⑥⑥⑥⑥⑥⑥⑥⑥  z  �  �❈  ❈  ❈  ❈  ❈  ❈  ❈  ❈  �  ⑥⑥⑥⑥⑥⑥⑥⑥ �❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄  � �❄  ❄  ❄  ❄  ❄  ❄  ❄  ❄  �  � � �  y  (2)  �  �  w  �  �  �⑤  ⑤  ⑤  ⑤  ⑤  ⑤  ⑤  ⑤ �  �⑧  ⑧  ⑧  ⑧  ⑧  ⑧  ⑧  ⑧ �  �  �  By construction the adjacency matrix M′ for the graph Λ′ satisfies  I − (M′)t =  �  B11  2B12  B21  B22 + B23  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  Therefore, we can use the graph Λ′ to find the cokernel and kernel of (∂1)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Using the same  logic and terminology we used when calculating the complex K-theory, we see that w is  a zero vertex because it emits an edge to a vertex v which supports a loop, and the edge  from w to v is the only non-loop edge which points to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Indeed, every vertex along the  bottom row of the graph Λ′ except y is a zero vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' The fact that these zero vertices  (with the exception of w) only receive edges from one (potentially) nonzero vertex on the  top row of Λ′ implies that every vertex in the top row except x and z are also zero vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  Now, w is a zero vertex, but since there are two edges from y to w we obtain the relation  [pw] = [pw] + 2[py] which implies that 2[py] = 0, but [py] ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Finally, from the relations  [px] = −[py] and [pz] = [py] we conclude that coker (∂1)0 = Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  To compute ker(∂1)0, we also proceed as in the computations for the complex case: All  of the vertices in the bottom row of Λ′, save w, are null vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Moreover, if a null vertex  v emits n edges to a single potentially non-null vertex u, we must have nα(u) = 0 for any  α ∈ NΛ′, and as α(u) ∈ Z we conclude that u must also be null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' It follows that w is null, as  are all of the vertices in the top row of Λ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' That is, ker(∂1)0 = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  Now, with the three rows that we’ve identified, the spectral sequence (8) implies that  KO0(C∗  R(Λ, γ)) ∼= KO1(C∗  R(Λ, γ)) ∼= Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' We claim that using this we can compute KOi(C∗  R(Λ, γ))  for 2 ≤ i ≤ 7 using the long exact sequence (5) and other aspects of CR-structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' The  fact that KU6(C∗  R(Λ, γ)) = KU0(C∗  R(Λ, γ)) = 0 implies that η0 is injective, and hence an  12  JEFFREY L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' BOERSEMA AND SARAH L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' BROWNE AND ELIZABETH GILLASPY  isomorphism, and also that η−1 is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Since KU2(C∗  R(Λ, γ)) = 0 it follows that η1 is  surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Thus if KO2(C∗  R(Λ, γ)) has a non-zero element, then it would have to be in the  image of η1 ◦ η0 ◦ η−1 = η3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' But η3 = 0 for all real C∗-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Thus KO2(C∗  R(Λ, γ)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  Since KO2(C∗  R(Λ, γ)) = 0, the long exact sequence implies that c3: KO3(C∗  R(Λ, γ)) →  KU3(C∗  R(Λ, γ)) = Z is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' This forces KO3(C∗  R(Λ, γ)) = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Moreover, as im r1 =  ker η1 = Z2 we must have r1 : Z → Z2 the unique nonzero map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Since im c3 ∼= ker r1, we  conclude that c3 is multiplication by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' The relation rc = 2 then implies that r3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  Continuing this process using the long exact sequence, we compute that KO∗(C∗  R(Λ, γ))) =  (Z2, Z2, 0, Z, 0, 0, 0, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' The module maps η, r, c, ψ are then completely determined by these  groups and the long exact sequence (5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' that is, K  CR(C∗  R(Λ, γ)) and hence K  CRT(C∗  R(Λ, γ))  coincide with ΣK  CRT(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  □  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Suppose that (Λ1, γ1) and (Λ2, γ2) are higher-rank graphs with involutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  Then (Λ1 × Λ2, γ1 × γ2) is a higher-rank graph with involution and  C∗  R(Λ1 × Λ2, γ1 × γ2) ∼= C∗  R(Λ1, γ1) ⊗R C∗  R(Λ2, γ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Assume that Λ1 and Λ2 have rank k1 and k2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  From [KP00, Proposi-  tion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='8] the product Λ1 × Λ2 is a graph of rank k1 + k2, with degree functor d(λ1, λ2) =  d(λ1) + d(λ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Furthermore, there is an involution γ on Λ1 × Λ2 defined by γ(λ1, λ2) =  (γ1(λ1), γ2(λ2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  From [KP00, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='5], there is an isomorphism φ: C∗(Λ1 × Λ2) → C∗(Λ1) ⊗ C∗(Λ2)  defined by φ(s(λ1,λ2)) = sλ1 ⊗sλ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' To finish the proof, we need only show that φ preserves the  real structures (4) of C∗(Λ1 × Λ2) and C∗(Λ1) ⊗ C∗(Λ2) which are induced by the graphical  involutions γi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' This is straightforward:  φ(�γ(s(λ1,λ2)) = φ(s∗  (γ1(λ1),γ2(λ2))))  = s∗  γ1(λ1) ⊗ s∗  γ2(λ2)  = �γ1(sλ1) ⊗ �γ2(sλ2)  = �  γ1 ⊗ γ2(sλ1 ⊗ sλ2)  = �  γ1 ⊗ γ2(φ(s(λ1,λ2))) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  □  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Let n be an odd integer, n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' There exists a rank-3 graph with invo-  lution (Λn, γn) such that C∗  R(Λn, γn) ∼= K  R ⊗R E  R  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  Furthermore, there exists a projection  p ∈ C∗  R(Λn, γn) such that pC∗  R(Λn, γn)p ∼= E  R  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Let (Λ, γ) be the 1-graph given by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='1 and let (En, γn) be the finite 1-  graph with involution from Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='2 of [Boe17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Then both C∗  R(Λ, γ) and C∗  R(En, γn) are  simple and purely infinite and we have  K  CR(C∗  R(Λ, γ)) ∼= ΣK  CR(R)  and  K  CR(C∗  R(En, γn)) ∼= Σ6K  CR(En) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  This implies by [BRS11, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='1] that  K  CRT(C∗  R(Λ, γ)) ∼= ΣK  CRT(R)  and  K  CRT(C∗  R(En, γn)) ∼= Σ6K  CRT(En) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  Let (Λn, γn) be the product rank-3 graph with involution  (Λn, γn) = (Λ, γ) × (Λ, γ) × (En, γn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  THE STABLE EXOTIC CUNTZ ALGEBRAS ARE HIGHER-RANK GRAPH ALGEBRAS  13  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='2 then implies that  C∗  R(Λn, γn) ∼= C∗  R(Λ, γ) ⊗R C∗  R(Λ, γ) ⊗R C∗  R(En, γn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  Now, K  CRT(C∗  R(Λ, γ)) is a free CRT -module, since it is isomorphic to a suspension of K  CRT(R)  (see [Boe02, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Therefore the K¨unneth formula for the K-theory of real C∗-  algebras (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='5 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='2 of [Boe02]) gives  K  CRT(C∗  R(Λn, γn)) ∼= K  CRT(C∗  R(Λ, γ)) ⊗CRT K  CRT(C∗  R(Λ, γ)) ⊗CRT K  CRT(C∗  R(En, γn))  ∼= Σ2K  CRT(R) ⊗CRT Σ6K  CRT(En)  ∼= K  CRT(En) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  Note that C∗  R(Λn, γn) is a stable, simple, purely infinite, real C∗-algebra, thanks to Propo-  sition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='1 and [Boe17, Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' We also know that KR ⊗R En is a a stable, simple,  purely infinite, real C∗-algebra, because its complexification K ⊗ On is simple and purely  infinite (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='9 of [BRS11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Thus the first statement of the theorem follows by  the classification of real Kirchberg algebras, [BRS11, Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='2, Part (1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  To prove the second statement, by [BRS11, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='13] there is a projection p ∈  C∗  R(Λn, γn) such that [p] is a generator of KO0(C∗  R(Λn, γn)) = Z2(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Then  K  CR(pC∗  R(Λn, γn)p) ∼= K  CR(C∗  R(Λn, γn)) ∼= K  CR(En)  (where the first isomorphism is by [Boe06, Proposition 9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Furthermore the class of the  identity [p] ∈ KO0(pC∗  R(Λ, γ)p) ∼= Z2(n−1) corresponds under this isomorphism to the class  of the identity [1] ∈ KO0(En) ∼= Z2(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Therefore by [BRS11, Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='2, Part (2)], we  have pC∗  R(Λ, γ)p ∼= En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Appendix – real Kirchberg suspension algebras  In the previous section, we introduced a graph with involution for which K  CR(C∗(Λ, γ)) ∼=  ΣK  CR(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' We consider this algebra as a sort of real Kirchberg suspension, since it is a real  purely infinite simple stable nuclear C∗-algebra satisfying the UCT, and with the same KK-  type as the suspension algebra SR ∼= C0((0, 1), R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' By repeatedly taking the product of this  graph with itself, which corresponds to repeatedly tensoring this algebra with itself, we can  obtain a higher-rank graph, the real C∗-algebra of which is a real Kirchberg algebra with the  same KK-type as SiR for any i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' These tensor products will be higher-rank graph algebras  of rank i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' It is natural to ask which of these suspensions can be obtained from a 1-graph  with involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' In this section, we will answer this question completely, providing a full  characterization of the integers i (mod 8) for which there exists a 1-graph with involution  (Λ, γ) such that K  CR(C∗(Λ, γ)) ∼= ΣiK  CR(R) ∼= K  CR(SiR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' For the positive results, we will  exhibit directly the appropriate graph or graph with involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' For each i = {−2, −1, 0, 1} there exists a 1-graph with involution (Λ, γ)  such that K  CR(C∗(Λ, γ)) ∼= ΣiK  CR(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Furthermore, C∗  R(Λ, γ) is simple and purely infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  Sketch of proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' For each i we show below a graph or graph with involution that satisfies  K  CR(C∗(Λ, γ)) ∼= ΣiK  CR(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' The K-theory calculations, not shown, are carried out using  the same techniques as in the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  14  JEFFREY L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' BOERSEMA AND SARAH L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' BROWNE AND ELIZABETH GILLASPY  i = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' The graph Λ is shown below;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' we equip it with the non-trivial involution γ which  interchanges the right-hand branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  �  ❅  ❅  ❅  ❅  ❅  ❅  ❅  �⑦⑦⑦⑦⑦⑦⑦  �  �  ❅  ❅  ❅  ❅  ❅  ❅  ❅  �⑦⑦⑦⑦⑦⑦⑦  �  �  ❅  ❅  ❅  ❅  ❅  ❅  ❅  �⑦⑦⑦⑦⑦⑦⑦  � �  �⑦⑦⑦⑦⑦⑦⑦  �  ❅  ❅  ❅  ❅  ❅  ❅  ❅ �  �⑦⑦⑦⑦⑦⑦⑦  �  ❅  ❅  ❅  ❅  ❅  ❅  ❅ �  �⑦⑦⑦⑦⑦⑦⑦  �  ❅  ❅  ❅  ❅  ❅  ❅  ❅  �  �  � •  � •  � •  �  �  � •  �  � � •  � •  �  �⑦  ⑦  ⑦  ⑦  ⑦  ⑦  ⑦  �  �❅  ❅  ❅  ❅  ❅  ❅  ❅ � •  � •  � •  �  � �  ⑦  ⑦  ⑦  ⑦  ⑦  ⑦  ⑦  �❅❅❅❅❅❅❅  � �  ⑦  ⑦  ⑦  ⑦  ⑦  ⑦  ⑦  �❅❅❅❅❅❅❅  � �  ⑦  ⑦  ⑦  ⑦  ⑦  ⑦  ⑦  �❅❅❅❅❅❅❅  �  i = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' The graph Λ is shown below, with trivial involution γ = id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' �❅  ❅  ❅  ❅  ❅  ❅  ❅  � �❅  ❅  ❅  ❅  ❅  ❅  ❅  � �❅  ❅  ❅  ❅  ❅  ❅  ❅  � �❅  ❅  ❅  ❅  ❅  ❅  ❅  � �⑦  ⑦  ⑦  ⑦  ⑦  ⑦  ⑦ �  �⑦  ⑦  ⑦  ⑦  ⑦  ⑦  ⑦ �  �⑦  ⑦  ⑦  ⑦  ⑦  ⑦  ⑦ �  �⑦  ⑦  ⑦  ⑦  ⑦  ⑦  ⑦ �  �  �  i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' The graph Λ is shown below, with trivial involution γ = id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' �⑦⑦⑦⑦⑦⑦⑦  �  �  ❅  ❅  ❅  ❅  ❅  ❅  ❅ �⑦⑦⑦⑦⑦⑦⑦  �  �  ❅  ❅  ❅  ❅  ❅  ❅  ❅ �⑦⑦⑦⑦⑦⑦⑦  �  �  ❅  ❅  ❅  ❅  ❅  ❅  ❅ �⑦⑦⑦⑦⑦⑦⑦  �  �  ❅  ❅  ❅  ❅  ❅  ❅  ❅ � •  � •  � •  � •  �  �  i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' The graph Λ is shown below with non-trivial involution γ, as in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
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+page_content='⑦  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='For the i = −2 graph,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' one can determine all of the groups KOi(C∗  R(Λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' γ)) from the  associated spectral sequence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' except for KO2(C∗  R(Λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' γ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' In that case, the spectral sequence  KO2(C∗  R(Λ, γ)) has the filtration 0 → Z → KO2(C∗  R(Λ, γ)) → Z2 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  Although this  filtration by itself does not determine KO2(C∗  R(Λ, γ)), the long exact sequence (5) forces  KO2(C∗  R(Λ, γ)) = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Moreover, the module maps r, c, η, ψ are uniquely determined by (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  □  THE STABLE EXOTIC CUNTZ ALGEBRAS ARE HIGHER-RANK GRAPH ALGEBRAS  15  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' For 2 ≤ i ≤ 5, there does not exist a 1-graph (Λ, γ) with involution such  that K  CR(C∗  R(Λ, γ)) ∼= ΣiK  CR(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Suppose that (Λ, γ) is a graph with involution and K  CR(C∗  R(Λ, γ)) ∼= ΣiK  CR(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  The real Pimsner-Voiculescu sequence (or equivalently, the real Evans spectral sequence)  for K  CR(C∗(Λ, γ)) implies that KO−1(C∗  R(Λ, γ)) and KO−3(C∗  R(Λ, γ)) are free abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  But recall that KO1(R) ∼= KO2(R) ∼= Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Thus the group (Σ2KO(R))−1 = KO1(R) has  torsion, implying that K  CR(C∗  R(Λ, γ)) ≇ Σ2KO  CR(R), hence i ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Similarly, the groups  (Σ3KO(R))−1, (Σ4KO(R))−3, and (Σ5KO(R))−3 have torsion, showing that i ̸= 3, 4, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  □  References  [BG22]  Jeffrey L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Boersema and Elizabeth Gillaspy, K-theory for real k-graph C∗-algebras, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' K-  Theory 7 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' 2, 395–440.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  [BHRS02]  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Bates, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
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+page_content=' Raeburn, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
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+page_content=' 4, 1159–1176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content='  [Boe02]  Jeffrey L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Boersema, Real C∗-algebras, united K-theory, and the K¨unneth formula, K-Theory  26 (2002), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
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+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
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+page_content='  [BRS11]  Jeffrey L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Boersema, Efren Ruiz, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Stacey, The classification of real purely infinite simple  C∗-algebras, Doc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' 16 (2011), 619–655.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' MR 2837543  [EFG+21] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Eckhardt, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Fieldhouse, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Gent, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Gillaspy, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
+page_content=' Gonzales, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'}
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