# Social and Behavioral Sciences Commons™

Faculty of Engineering and Information Sciences - Papers: Part A

Algebras

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## Full-Text Articles in Social and Behavioral Sciences

Equivalence And Stable Isomorphism Of Groupoids, And Diagonal-Preserving Stable Isomorphisms Of Graph C*-Algebras And Leavitt Path Algebras, Toke Meier Carlsen, Efren Ruiz, Aidan Sims Jan 2017

#### Equivalence And Stable Isomorphism Of Groupoids, And Diagonal-Preserving Stable Isomorphisms Of Graph C*-Algebras And Leavitt Path Algebras, Toke Meier Carlsen, Efren Ruiz, Aidan Sims

##### Faculty of Engineering and Information Sciences - Papers: Part A

We prove that ample groupoids with σ-compact unit spaces are equivalent if and only if they are stably isomorphic in an appropriate sense, and relate this to Matui's notion of Kakutani equivalence. We use this result to show that diagonal-preserving stable isomorphisms of graph C*-algebras or Leavitt path algebras give rise to isomorphisms of the groupoids of the associated stabilised graphs. We deduce that the Leavitt path algebras LZ(E2) and LZ(E2-) are not stably *-isomorphic.

The Extension Class And Kms States For Cuntz-Pimsner Algebras Of Some Bi-Hilbertian Bimodules, Adam C. Rennie, David I. Robertson, Aidan Sims Jan 2017

#### The Extension Class And Kms States For Cuntz-Pimsner Algebras Of Some Bi-Hilbertian Bimodules, Adam C. Rennie, David I. Robertson, Aidan Sims

##### Faculty of Engineering and Information Sciences - Papers: Part A

For bi-Hilbertian A-bimodules, in the sense of Kajiwara-Pinzari-Watatani, we construct a Kasparov module representing the extension class defining the Cuntz-Pimsner algebra. The construction utilises a singular expectation which is defined using the C*-module version of the Jones index for bi-Hilbertian bimodules. The Jones index data also determines a novel quasi-free dynamics and KMS states on these Cuntz-Pimsner algebras.

Cartan Subalgebras In C*-Algebras Of Hausdorff Étale Groupoids, Jonathan H. Brown, Gabriel Nagy, Sarah Reznikoff, Aidan Sims, Dana P. Williams Jan 2016

#### Cartan Subalgebras In C*-Algebras Of Hausdorff Étale Groupoids, Jonathan H. Brown, Gabriel Nagy, Sarah Reznikoff, Aidan Sims, Dana P. Williams

##### Faculty of Engineering and Information Sciences - Papers: Part A

The reduced C*-algebra of the interior of the isotropy in any Hausdorff étale groupoid G embeds as a C*-subalgebra M of the reduced C*-algebra of G. We prove that the set of pure states of M with unique extension is dense, and deduce that any representation of the reduced C*-algebra of G that is injective on M is faithful. We prove that there is a conditional expectation from the reduced C*-algebra of G onto M if and only if the interior of the isotropy in G is closed. Using this, we prove that when the interior of the isotropy is …

Af-Embeddability Of 2-Graph Algebras And Quasidiagonality Of K-Graph Algebras, Lisa Orloff Clark, Astrid An Huef, Aidan Sims Jan 2016

#### Af-Embeddability Of 2-Graph Algebras And Quasidiagonality Of K-Graph Algebras, Lisa Orloff Clark, Astrid An Huef, Aidan Sims

##### Faculty of Engineering and Information Sciences - Papers: Part A

We characterise quasidiagonality of the C*-algebra of a cofinal k-graph in terms of an algebraic condition involving the coordinate matrices of the graph. This result covers all simple k-graph C*-algebras. In the special case of cofinal 2-graphs we further prove that AF-embeddability, quasidiagonality and stable finiteness of the 2-graph algebra are all equivalent.

Uct-Kirchberg Algebras Have Nuclear Dimension One, Efren Ruiz, Aidan Sims, Adam P. Sorensen Jan 2015

#### Uct-Kirchberg Algebras Have Nuclear Dimension One, Efren Ruiz, Aidan Sims, Adam P. Sorensen

##### Faculty of Engineering and Information Sciences - Papers: Part A

We prove that every Kirchberg algebra in the UCT class has nuclear dimension 1. We first show that Kirchberg 2-graph algebras with trivial K0 and finite K1 have nuclear dimension 1 by adapting a technique developed by Winter and Zacharias for Cuntz algebras. We then prove that every Kirchberg algebra in the UCT class is a direct limit of 2-graph algebras to obtain our main theorem.

The Nuclear Dimension Of Graph C*-Algebras, Efren Ruiz, Aidan Sims, Mark Tomforde Jan 2015

#### The Nuclear Dimension Of Graph C*-Algebras, Efren Ruiz, Aidan Sims, Mark Tomforde

##### Faculty of Engineering and Information Sciences - Papers: Part A

Consider a graph C⁎C⁎-algebra C⁎(E)C⁎(E) with a purely infinite ideal I (possibly all of C⁎(E)C⁎(E)) such that I has only finitely many ideals and C⁎(E)/IC⁎(E)/I is approximately finite dimensional. We prove that the nuclear dimension of C⁎(E)C⁎(E) is 1. If I has infinitely many ideals, then the nuclear dimension of C⁎(E)C⁎(E) is either 1 or 2.

Twisted K-Graph Algebras Associated To Bratteli Diagrams, David Pask, Adam Sierakowski, Aidan Sims Jan 2015

#### Twisted K-Graph Algebras Associated To Bratteli Diagrams, David Pask, Adam Sierakowski, Aidan Sims

##### Faculty of Engineering and Information Sciences - Papers: Part A

2015 Springer Basel Given a system of coverings of k-graphs, we show that the second cohomology of the resulting (k + 1)-graph is isomorphic to that of any one of the k-graphs in the system, and compute the semifinite traces of the resulting twisted (k + 1)-graph C*-algebras. We then consider Bratteli diagrams of 2-graphs whose twisted C*-algebras are matrix algebras over noncommutative tori. For such systems we calculate the ordered K-theory of the resulting twisted 3-graph C*-algebras. We deduce that every such C*-algebra is Morita equivalent to the C*-algebra of a rank-2 Bratteli diagram in the sense of Pask-Raeburn-Rørdam-Sims.

Von Neumann Algebras Of Strongly Connected Higher-Rank Graphs, Marcelo Laca, Nadia S. Larsen, Sergey Neshveyev, Aidan Sims, Samuel B. Webster Jan 2015

#### Von Neumann Algebras Of Strongly Connected Higher-Rank Graphs, Marcelo Laca, Nadia S. Larsen, Sergey Neshveyev, Aidan Sims, Samuel B. Webster

##### Faculty of Engineering and Information Sciences - Papers: Part A

We investigate the factor types of the extremal KMS states for the preferred dynamics on the Toeplitz algebra and the Cuntz-Krieger algebra of a strongly connected finite (Formula presented.)-graph. For inverse temperatures above 1, all of the extremal KMS states are of type (Formula presented.). At inverse temperature 1, there is a dichotomy: if the (Formula presented.)-graph is a simple (Formula presented.)-dimensional cycle, we obtain a finite type (Formula presented.) factor; otherwise we obtain a type III factor, whose Connes invariant we compute in terms of the spectral radii of the coordinate matrices and the degrees of cycles in the …

Group Actions On Labeled Graphs And Their C*-Algebras, Teresa Bates, David Pask, Paulette Willis Jan 2014

#### Group Actions On Labeled Graphs And Their C*-Algebras, Teresa Bates, David Pask, Paulette Willis

##### Faculty of Engineering and Information Sciences - Papers: Part A

We introduce the notion of the action of a group on a labeled graph and the quotient object, also a labeled graph. We define a skew product labeled graph and use it to prove a version of the Gross–Tucker theorem for labeled graphs. We then apply these results to the C -algebra associated to a labeled graph and provide some applications in non-Abelian duality.

Omitting Types And Af Algebras, Kevin Carlson, Enoch Cheung, Ilijas Farah, Alexander Gerhardt-Bourke, Bradd Hart, Leanne Mezuman, Nigel Sequeira, Alexander Sherman Jan 2014

#### Omitting Types And Af Algebras, Kevin Carlson, Enoch Cheung, Ilijas Farah, Alexander Gerhardt-Bourke, Bradd Hart, Leanne Mezuman, Nigel Sequeira, Alexander Sherman

##### Faculty of Engineering and Information Sciences - Papers: Part A

We prove that the classes of UHF algebras and AF algebras, while not axiomatizable, can be characterized as those C*-algebras that omit certain types in the logic of metric structures.

Purely Infinite C*-Algebras Associated To Etale Groupoids, Jonathon Brown, Les Clark, Adam Sierakowski Jan 2014

#### Purely Infinite C*-Algebras Associated To Etale Groupoids, Jonathon Brown, Les Clark, Adam Sierakowski

##### Faculty of Engineering and Information Sciences - Papers: Part A

No abstract provided.

Twisted C-Algebras Associated To Finitely Aligned Higher-Rank Graphs, Aidan Sims, Benjamin Whitehead, Michael Whittaker Jan 2014

#### Twisted C-Algebras Associated To Finitely Aligned Higher-Rank Graphs, Aidan Sims, Benjamin Whitehead, Michael Whittaker

##### Faculty of Engineering and Information Sciences - Papers: Part A

We introduce twisted relative Cuntz-Krieger algebras associated to finitely aligned higher-rank graphs and give a comprehensive treatment of their fundamental structural properties. We establish versions of the usual uniqueness theorems and the classification of gauge-invariant ideals. We show that all twisted relative Cuntz- Krieger algebras associated to finitely aligned higher-rank graphs are nuclear and satisfy the UCT, and that for twists that lift to real-valued cocycles, the K-theory of a twisted relative Cuntz-Krieger algebra is independent of the twist. In the final section, we identify a sufficient condition for simplicity of twisted Cuntz-Krieger algebras associated to higher-rank graphs which are …

Kms States On The C-Algebras Of Reducible Graphs, Astrid An Huef, Marcelo Laca, Iain Raeburn, Aidan Sims Jan 2014

#### Kms States On The C-Algebras Of Reducible Graphs, Astrid An Huef, Marcelo Laca, Iain Raeburn, Aidan Sims

##### Faculty of Engineering and Information Sciences - Papers: Part A

We consider the dynamics on the C-algebras of finite graphs obtained by lifting the gauge action to an action of the real line. Enomoto, Fujii and Watatani [KMS states for gauge action on OA. Math. Japon. 29 (1984), 607-619] proved that if the vertex matrix of the graph is irreducible, then the dynamics on the graph algebra admits a single Kubo-Martin-Schwinger (KMS) state. We have previously studied the dynamics on the Toeplitz algebra, and explicitly described a finite-dimensional simplex of KMS states for inverse temperatures above a critical value. Here we study the KMS states for graphs with reducible vertex …

Simplicity Of Algebras Associated To Étale Groupoids, Jonathan Brown, Lisa Orloff Clark, Cynthia Farthing, Aidan Sims Jan 2014

#### Simplicity Of Algebras Associated To Étale Groupoids, Jonathan Brown, Lisa Orloff Clark, Cynthia Farthing, Aidan Sims

##### Faculty of Engineering and Information Sciences - Papers: Part A

We prove that the full C*-algebra of a second-countable, Hausdorff, etale, amenable groupoid is simple if and only if the groupoid is both topologically principal and minimal. We also show that if G has totally disconnected unit space, then the complex *-algebra of its inverse semigroup of compact open bisections, as introduced by Steinberg, is simple if and only if G is both effective and minimal.

Equilibrium States On The Cuntz-Pimsner Algebras Of Self-Similar Actions, Marcelo Laca, Iain Raeburn, Jacqui Ramagge, Michael Whittaker Jan 2014

#### Equilibrium States On The Cuntz-Pimsner Algebras Of Self-Similar Actions, Marcelo Laca, Iain Raeburn, Jacqui Ramagge, Michael Whittaker

##### Faculty of Engineering and Information Sciences - Papers: Part A

We consider a family of Cuntz-Pimsner algebras associated to self-similar group actions, and their Toeplitz analogues. Both families carry natural dynamics implemented by automorphic actions of the real line, and we investigate the equilibrium states (the KMS states) for these dynamical systems. We find that for all inverse temperatures above a critical value, the KMS states on the Toeplitz algebra are given, in a very concrete way, by traces on the full group algebra of the group. At the critical inverse temperature, the KMS states factor through states of the Cuntz-Pimsner algebra; if the self-similar group is contracting, then the …

An Elementary Approach To C*-Algebras Associated To Topological Graphs, Hui Li, David Pask, Aidan Sims Jan 2014

#### An Elementary Approach To C*-Algebras Associated To Topological Graphs, Hui Li, David Pask, Aidan Sims

##### Faculty of Engineering and Information Sciences - Papers: Part A

We develop notions of a representation of a toopological grapph E and of a covariant representation of a topological graph E which do onot require the machinery of C* -correspondences and Cuntz-Pimsner alegebars. We show that the C* -algebra generated by a universal representation of E is isomorphic to the Toeplitz algebra of Katsura's topological-graph bimodule, and that the C* palgebra generated by a universal covariant representation of E is isomorphic to Katsura's topological graph C* -algebra. We exhibit our resluts by constructing the isomorphism between the C* -algebra of the row-finite directed graph E with no sources and the …

Kms States On C*-Algebras Associated To Higher-Rank Graphs, Astrid An Huef, Marcelo Laca, Iain Raeburn, Aidan Sims Jan 2014

#### Kms States On C*-Algebras Associated To Higher-Rank Graphs, Astrid An Huef, Marcelo Laca, Iain Raeburn, Aidan Sims

##### Faculty of Engineering and Information Sciences - Papers: Part A

Consider a higher rank graph of rank k. Both the Cuntz-Krieger algebra and Toeplitz-Cuntz-Krieger algebra of the graph carry natural gauge actions of the torus Tk, and restricting these guage actions to one parameter subgroups of Tk gives dynamical systems involving actions of the real line. We study the KMS states of these dynamical systems. We find that for large inverse temperatures B, the simplex of KMS B states of the Toeplitz-Cuntz-Krieger algebra has dimension d one less than the number of vertices in the graph. We also show that there is a preferred dynamics for which there is a …

Zappa-Szep Products Of Semigroups And Their C*-Algebras, Nathan D. Brownlowe, Jacqueline Ramagge, David I. Robertson, Michael F. Whittaker Jan 2014

#### Zappa-Szep Products Of Semigroups And Their C*-Algebras, Nathan D. Brownlowe, Jacqueline Ramagge, David I. Robertson, Michael F. Whittaker

##### Faculty of Engineering and Information Sciences - Papers: Part A

Zappa-Szep products of semigroups provide a rich class of examples of semigroups that include the self-similar group actions of Nekrashevych. We use Li's construction of semigroups C*-algebras to associate a C*-algebra to Zappa-Szep products and give an explicit presentation of the algebra. We then define a quotient C*-algebra that generalises the Cuntz-Pimsner algebras for self-similar actions. We indicate how knowne examples, previously viewed as distinct classes, fit into our unifying framework. We specifically discuss the Baumslag-Solitar groups, the binary adding machine, the semigroup NXNx, and the ax+b semigroup ZXZx.

On The K-Theory Of Twisted Higher-Rank-Graph C*-Algebras, Alex Kumjian, David Pask, Aidan Sims Jan 2013

#### On The K-Theory Of Twisted Higher-Rank-Graph C*-Algebras, Alex Kumjian, David Pask, Aidan Sims

##### Faculty of Engineering and Information Sciences - Papers: Part A

We investigate the K-theory of twisted higher-rank-graph algebras by adapting parts of Elliott's computation of the K-theory of the rotation algebras. We show that each 2-cocycle on a higher-rank graph taking values in an abelian group determines a continuous bundle of twisted higher-rank graph algebras over the dual group. We use this to show that for a circle-valued 2-cocycle on a higher-rank graph obtained by exponentiating a real-valued cocycle, the K-theory of the twisted higher-rank graph algebra coincides with that of the untwisted one.

Kms States On The C*-Algebras Of Finite Graphs, Astrid An Huef, Marcelo Laca, Iain F. Raeburn, Aidan D. Sims Jan 2013

#### Kms States On The C*-Algebras Of Finite Graphs, Astrid An Huef, Marcelo Laca, Iain F. Raeburn, Aidan D. Sims

##### Faculty of Engineering and Information Sciences - Papers: Part A

We consider a finite directed graph E, and the gauge action on its Toeplitz-Cuntz-Krieger algebra, viewed as an action of R. For inverse temperatures larger than a critical value βc, we give an explicit construction of all the KMSβ states. If the graph is strongly connected, then there is a unique KMSβc state, and this state factors through the quotient map onto C*(E). Our approach is direct and relatively elementary.

Remarks On Some Fundamental Results About Higher-Rank Graphs And Their C*-Algebras, Robert Hazlewood, Iain Raeburn, Aidan Sims, Samuel B. G Webster Jan 2013

#### Remarks On Some Fundamental Results About Higher-Rank Graphs And Their C*-Algebras, Robert Hazlewood, Iain Raeburn, Aidan Sims, Samuel B. G Webster

##### Faculty of Engineering and Information Sciences - Papers: Part A

Results of Fowler and Sims show that every k-graph is completely determined by its k-coloured skeleton and collection of commuting squares. Here we give an explicit description of the k-graph associated with a given skeleton and collection of squares and show that two k-graphs are isomorphic if and only if there is an isomorphism of their skeletons which preserves commuting squares. We use this to prove directly that each k-graph. is isomorphic to the quotient of the path category of its skeleton by the equivalence relation determined by the commuting squares, and show that this extends to a homeomorphism of …

Purely Infinite C-Algebras Arising From Crossed Products, Mikael Rordam, Adam Sierakowski Jan 2012

#### Purely Infinite C-Algebras Arising From Crossed Products, Mikael Rordam, Adam Sierakowski

##### Faculty of Engineering and Information Sciences - Papers: Part A

We study conditions that will ensure that a crossed product of a C-algebra by a discrete exact group is purely infinite (simple or non-simple). We are particularly interested in the case of a discrete non-amenable exact group acting on a commutative C-algebra, where our sufficient conditions can be phrased in terms of paradoxicality of subsets of the spectrum of the abelian C-algebra. As an application of our results we show that every discrete countable non-amenable exact group admits a free amenable minimal action on the Cantor set such that the corresponding crossed product C-algebra is a Kirchberg algebra in the …

C*-Algebras Associated To C*-Correspondences And Applications To Mirror Quantum Spheres, David I. Robertson, Wojciech Szymanski Jan 2011

#### C*-Algebras Associated To C*-Correspondences And Applications To Mirror Quantum Spheres, David I. Robertson, Wojciech Szymanski

##### Faculty of Engineering and Information Sciences - Papers: Part A

The structure of the C*-algebras corresponding to even-dimensional mirror quantum spheres is investigated. It is shown that they are isomorphic to both Cuntz-Pimsner algebras of certain C*-correspondences and C*-algebras of certain labelled graphs. In order to achieve this, categories of labelled graphs and C*-correspondences are studied. A functor from labelled graphs to C*-correspondences is constructed, such that the corresponding associated C*-algebras are isomorphic. Furthermore, it is shown that C*-correspondences for the mirror quantum spheres arise via a general construction of restricted direct sum.

Families Of Type Iii Kms States On A Class Of C-Algebras Containing On And Qn, A L. Carey, J Phillips, I F. Putnam, A Rennie Jan 2011

#### Families Of Type Iii Kms States On A Class Of C-Algebras Containing On And Qn, A L. Carey, J Phillips, I F. Putnam, A Rennie

##### Faculty of Engineering and Information Sciences - Papers: Part A

We construct a family of purely infinite C¤-algebras, Q¸ for ¸ 2 (0, 1) that are classified by their K-groups. There is an action of the circle T with a unique KMS state Ã on each Q¸. For ¸ = 1/n, Q1/n »= On, with its usual T action and KMS state. For ¸ = p/q, rational in lowest terms, Q¸ »= On (n = q − p + 1) with UHF fixed point algebra of type (pq)1. For any n > 1, Q¸ »= On for infinitely many ¸ with distinct KMS states and UHF fixed-point algebras. For any ¸ …

Purely Infinite Simple C*-Algebras Associated To Integer Dilation Matrices, Ruy Exel, Astrid An Huef, Iain Raeburn Jan 2011

#### Purely Infinite Simple C*-Algebras Associated To Integer Dilation Matrices, Ruy Exel, Astrid An Huef, Iain Raeburn

##### Faculty of Engineering and Information Sciences - Papers: Part A

Given an n×n integer matrix A whose eigenvalues are strictly greater than 1 in absolute value, let σA be the transformation of the n-torus Tn = Rn/Zn defined by σA(e2πix) = e2πiAx for x ∈ Rn. We study the associated crossed-product C∗-algebra, which is defined using a certain transfer operator for σA, proving it to be simple and purely infinite and computing its K-theory groups.

Semifinite Spectral Triples Associated With Graph C*-Algebras, Alan L. Carey, John Phillips, Adam Rennie Jan 2008

#### Semifinite Spectral Triples Associated With Graph C*-Algebras, Alan L. Carey, John Phillips, Adam Rennie

##### Faculty of Engineering and Information Sciences - Papers: Part A

We review the recent construction of semifinite spectral triples for graph C^*-algebras. These examples have inspired many other developments and we review some of these such as the relation between the semifinite index and the Kasparov product, examples of noncommutative manifolds, and an index theorem in twisted cyclic theory using a KMS state.

Extension Problems And Non-Abelian Duality For C*-Algebras, Astrid An Huef, S Kaliszewski, Iain Raeburn Jan 2007

#### Extension Problems And Non-Abelian Duality For C*-Algebras, Astrid An Huef, S Kaliszewski, Iain Raeburn

##### Faculty of Engineering and Information Sciences - Papers: Part A

Suppose that H is a closed subgroup of a locally compact group G. We show that a unitary representation U of H is the restriction of a unitary representation of G if and only if a dual representation Û of a crossed product C*(G) (G/H) is regular in an appropriate sense. We then discuss the problem of deciding whether a given representation is regular; we believe that this problem will prove to be an interesting test question in non-Abelian duality for crossed products of C*-algebras.

Properties Preserved Under Morita Equivalence Of C*-Algebras, Astrid An Huef, Iain Raeburn, Dana Williams Jan 2007

#### Properties Preserved Under Morita Equivalence Of C*-Algebras, Astrid An Huef, Iain Raeburn, Dana Williams

##### Faculty of Engineering and Information Sciences - Papers: Part A

We show that important structural properties of C*-algebras and the muliplicity numbers of representations are preserved under Morita equivalence.

Kk-Theory And Spectral Flow In Von Neumann Algebras, J Kaad, R Nest, Adam C. Rennie Jan 2007

#### Kk-Theory And Spectral Flow In Von Neumann Algebras, J Kaad, R Nest, Adam C. Rennie

##### Faculty of Engineering and Information Sciences - Papers: Part A

We present a definition of spectral flow for any norm closed ideal J in any von Neumann algebra N. Given a path of selfadjoint operators in N which are invertible in N/J, the spectral flow produces a class in Ko(J).Given a semifinite spectral triple (A, H, D) relative to (N, t) with A separable, we construct a class [D] ? KK1(A, K(N)). For a unitary u ? A, the von Neumann spectral flow between D and u*Du is equal to the Kasparov product [u]A[D], and is simply related to the numerical spectral flow, and a refined C*-spectral flow.

Exel's Crossed Product And Relative Cuntz-Pimsner Algebras, Nathan Brownlowe, Iain Raeburn Jan 2006

#### Exel's Crossed Product And Relative Cuntz-Pimsner Algebras, Nathan Brownlowe, Iain Raeburn

##### Faculty of Engineering and Information Sciences - Papers: Part A

We consider Exel's new construction of a crossed product of a $C^*$-algebra $A$ by an endomorphism $\alpha$. We prove that this crossed product is universal for an appropriate family of covariant representations, and we show that it can be realised as a relative Cuntz-Pimsner algbera. We describe a necessary and sufficient condition for the canonical map from $A$ into the crossed product to be injective, and present several examples to demonstrate the scope of this result. We also prove a gauge-invariant uniqueness theorem for the crossed product.