General Topology-I

Tables of integrals

Fundamental facts Basic theory: Nuclear and exact $\textrm{C}^*$-algebras: Definitions, basic facts and examples Tensor products Constructions Exact groups and related topics Amenable traces and Kirchberg's factorization property Quasidiagonal C*-algebras AF embeddablity Local reflexivity and other tensor product conditions Summary and open problems Special topics: Simple $\textrm{C}^*$-algebras Approximation properties for groups Weak expectation property and local lifting property Weakly exact von Neumann algebras Applications: Classification of group von Neumann algebras Herrero's approximation problem Counterexamples in $\textrm{K}$-homology and $\textrm{K}$-theory Appendices: Ultrafilters and ultraproducts Operator spaces, completely bounded maps and duality Lifting theorems Positive definite functions, cocycles and Schoenberg's Theorem Groups and graphs Bimodules over von Neumann algebras Bibliography Notation index Subject index.

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C*-Algebras and Finite-Dimensional Approximations

Operators on Hilbert Space.- C*-Algebras.- Von Neumann Algebras.- Further Structure.- K-Theory and Finiteness.

Operator Algebras: Theory of C*-Algebras and von Neumann Algebras

Graphs, Groupoids, and Cuntz–Krieger Algebras

We associate to each row-nite directed graph E a universal Cuntz-Krieger C-algebra C(E), and study how the distribution of loops in E aects the structure of C(E) .W e prove that C(E) is AF if and only if E has no loops. We describe an exit condition (L) on loops in E which allows us to prove an analogue of the Cuntz-Krieger uniqueness theorem and give a characterisation of when C(E) is purely innite. If the graph E satises (L) and is conal, then we have a dichotomy: if E has no loops, then C(E) is AF; if E has a loop, then C(E) is purely innite. If A is an n n f0; 1g-matrix, a Cuntz-Krieger A-family consists of n partial isometries Si on Hilbert space satisfying

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Cuntz-krieger algebras of directed graphs

Building on recent work of Robertson and Steger, we associate a C{algebra to a combinatorial object which may be thought of as a higher rank graph. This C{algebra is shown to be isomorphic to that of the associ- ated path groupoid. Various results in this paper give sucient conditions on the higher rank graph for the associated C{algebra to be: simple, purely in- nite and AF. Results concerning the structure of crossed products by certain natural actions of discrete groups are obtained; a technique for constructing rank 2 graphs from \commuting" rank 1 graphs is given.

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Higher Rank Graph C-Algebras

NSKI Abstract. We prove versions of the fundamental theorems about Cuntz-Krieger algebras for the C -algebras of row-finite graphs: directed graphs in which each vertex emits at most finitely many edges. Special cases of these results have previously been obtained using various powerful machines; our main point is that direct methods yield sharper results more easily. In the last few years various authors have considered analogues of the Cuntz-Krieger algebras associated to infinite directed graphs. In (12) and (11) these graph C -algebras were studied using a groupoid model and the deep results of Renault on the ideal structure of groupoid C -algebras; in (16) and (10) they were viewed as the Cuntz-Pimsner algebras of appropriate Hilbert bimodules, as introduced in (15). Because of the technical requirements of these general theories, it has usually been assumed that the graphs are locally finite, in the sense that every vertex receives and emits at most finitely many edges, and that the graphs do not have sinks. However, it was pointed out in (11) that to make sense of the Cuntz-Krieger relations in a C -algebra, one merely needs to insist that the graph is row-finite: each vertex emits at most finitely many edges. Here we shall prove versions of the fundamental theorems about Cuntz-Krieger algebras for the C -algebras of row-finite graphs, and use them to give a new description of the primitive ideal spaces of graph C -algebras. We prove a uniqueness theorem like that of (7) whenever every loop has an exit (11), and find a parametrisation of the ideals like that of (6) and (9) whenever the graph satisfies Condition (K) of (12): every vertex lies on either no loops or at least two loops. Both theorems apply to graphs with sinks; this new generality is important because it has been shown in (18) that the Cuntz-Krieger algebras of all infinite graphs and matrices can be approximated by the algebras of finite graphs with sinks. Our description of the primitive ideal space applies to any row-finite graph satisfying Condition (K).

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The c -algebras of row-finite graphs

Given a free action of a group is a certain zero-dimensional space canonically associated to the tree.

David Pask

Papers

Graphs, Groupoids, and Cuntz–Krieger Algebras

Cuntz-krieger algebras of directed graphs

Higher Rank Graph C-Algebras

The c -algebras of row-finite graphs

$C^*$-algebras of directed graphs and group actions