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Homology For Higher-Rank Graphs And Twisted C*-Algebras, Alex Kumjian, David Pask, Aidan Sims
Homology For Higher-Rank Graphs And Twisted C*-Algebras, Alex Kumjian, David Pask, Aidan Sims
Faculty of Informatics - Papers (Archive)
We introduce a homology theory for k-graphs and explore its fundamental properties. We establish connections with algebraic topology by showing that the homology of a k-graph coincides with the homology of its topological realisation as described by Kaliszewski et al. We exhibit combinatorial versions of a number of standard topological constructions, and show that they are compatible, from a homological point of view, with their topological counterparts. We show how to twist the C*-algebra of a k-graph by a T-valued 2-cocycle and demonstrate that examples include all noncommutative tori. In the appendices, we construct a cubical set …
Exel's Crossed Product For Non-Unital C*-Algebras, Nathan D. Brownlowe, Iain F. Raeburn, Sean T. Vittadello
Exel's Crossed Product For Non-Unital C*-Algebras, Nathan D. Brownlowe, Iain F. Raeburn, Sean T. Vittadello
Faculty of Informatics - Papers (Archive)
We consider a family of dynamical systems (A, alpha, L) in which a is an endomorphism of a C*-algebra A and L is a transfer operator for a. We extend Exel's construction of a crossed product to cover non-unital algebras A, and show that the C*-algebra of a locally finite graph can be realised as one of these crossed products. When A is commutative, we find criteria for the simplicity of the crossed product, and analyse the ideal structure of the crossed product.
C*-Algebras Of Tilings With Infinite Rotational Symmetry, Michael F. Whittaker
C*-Algebras Of Tilings With Infinite Rotational Symmetry, Michael F. Whittaker
Faculty of Informatics - Papers (Archive)
A tiling with infinite rotational symmetry, such as the Conway– Radin Pinwheel Tiling, gives rise to a topological dynamical system to which an etale equivalence relation is associated. A groupoid C¤-algebra for a tiling is produced and a separating dense set is exhibited in the C*-algebra which encodes the structure of the topological dynamical system. In the case of a substitution tiling, natural subsets of this separating dense set are used to define an AT-subalgebra of the C*-algebra. Finally our results are applied to the Pinwheel Tiling.
A Direct Approach To Co-Universal Algebras Associated To Directed Graphs, Aidan Sims, S B. Webster
A Direct Approach To Co-Universal Algebras Associated To Directed Graphs, Aidan Sims, S B. Webster
Faculty of Informatics - Papers (Archive)
We prove directly that if $E$ is a directed graph in which every cycle has an entrance, then there exists a $C^*$-algebra which is co-universal for Toeplitz-Cuntz-Krieger $E$-families. In particular, our proof does not invoke ideal-structure theory for graph algebras, nor does it involve use of the gauge action or its fixed point algebra.
Simplicity Of C*-Algebras Associated To Row-Finite Locally Convex Higher-Rank Graphs, David Robertson, Aidan Sims
Simplicity Of C*-Algebras Associated To Row-Finite Locally Convex Higher-Rank Graphs, David Robertson, Aidan Sims
Faculty of Informatics - Papers (Archive)
In a previous work, the authors showed that the C*-algebra C*(\Lambda) of a row-finite higher-rank graph \Lambda with no sources is simple if and only if \Lambda is both cofinal and aperiodic. In this paper, we generalise this result to row-finite higher-rank graphs which are locally convex (but may contain sources). Our main tool is Farthing's "removing sources" construction which embeds a row-finite locally convex higher-rank graph in a row-finite higher-rank graph with no sources in such a way that the associated C*-algebras are Morita equivalent.
Crossed Products Of K-Graph C*-Algebras By Zl, Cyntyhia Farthing, David Pask, Aidan Sims
Crossed Products Of K-Graph C*-Algebras By Zl, Cyntyhia Farthing, David Pask, Aidan Sims
Faculty of Informatics - Papers (Archive)
An action of Zl by automorphisms of a k-graph induces an action of Zl by automorphisms of the corresponding k-graph C*-algebra. We show how to construct a (k + l)-graph whose C*-algebra coincides with the crossed product of the original k-graph C)-algebra by Zl. We then investigate the structure of the crossed-product C*-algebra.
Group Extensions And The Primitive Ideal Spaces Of Toeplitz Algebras, Sriwulan Adji, Iain F. Raeburn, Rizky Rosjanuardi
Group Extensions And The Primitive Ideal Spaces Of Toeplitz Algebras, Sriwulan Adji, Iain F. Raeburn, Rizky Rosjanuardi
Faculty of Informatics - Papers (Archive)
Let Γ be a totally ordered abelian group and I an order ideal in Γ. We prove a theorem which relates the structure of the Toeplitz algebra T(Γ) to the structure of the Toeplitz algebras T(I) and T(Γ/I). We then describe the primitive ideal space of the Toeplitz algebra T(Γ) when the set Σ(Γ) of order ideals in Γ is well-ordered, and use this together with our structure theorem to deduce information about the ideal structure of T(Γ) when 0→ I→ Γ→ Γ/I→ 0 is a non-trivial group …
A Compactly Generated Group Whose Hecke Algebras Admit No Bounds On Their Representations, Udo Baumgartner, Jacqueline Ramagge, George A. Willis
A Compactly Generated Group Whose Hecke Algebras Admit No Bounds On Their Representations, Udo Baumgartner, Jacqueline Ramagge, George A. Willis
Faculty of Informatics - Papers (Archive)
We construct a compactly generated, totally disconnected, locally compact group whose Hecke algebra with respect to any compact open subgroup does not have a C∗-enveloping algebra. 2000 Mathematics Subject Classification. 20C08.
Flow Equivalence Of Graph Algebras, David A. Pask, Teresa G. Bates
Flow Equivalence Of Graph Algebras, David A. Pask, Teresa G. Bates
Faculty of Informatics - Papers (Archive)
This paper explores the effect of various graphical constructions upon the associated graph C∗-algebras. The graphical constructions in question arise naturally in the study of flow equivalence for topological Markov chains. We prove that out-splittings give rise to isomorphic graph algebras, and in-splittings give rise to strongly Morita equivalent C∗-algebras. We generalize the notion of a delay as defined in (D. Drinen, Preprint, Dartmouth College, 2001) to form in-delays and out-delays. We prove that these constructions give rise to Morita equivalent graph C∗-algebras. We provide examples which suggest that our results are the most general possible in the setting of …
Posets And Differential Graded Algebras, Jacqueline Ramagge, Wayne W. Wheeler
Posets And Differential Graded Algebras, Jacqueline Ramagge, Wayne W. Wheeler
Faculty of Informatics - Papers (Archive)
If P is a partially ordered set and R is a commutative ring, then a certain differential graded /f-algebra A,(P) is defined from the order relation on P. The algebra A.(Vi) corresponding to the empty poset is always contained in A.(P) so that A,(P) can be regarded as an /4.(0)-algebra. The main result of this paper shows that if R is an integral domain and P and P' are finite posets such that A.(P) = A.(P') as differential graded /4,(0)-algebras, then P and P' are isomorphic. 1991 Mathematics subject classification (Amer. Math. Soc): primary 06A06.