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11
Covering dimension for nuclear C ∗ algebras
, 2003
"... We introduce the completely positive rank, a notion of covering dimension for nuclear C ∗algebras and analyze some of its properties. The completely positive rank behaves nicely with respect to direct sums, quotients, ideals and inductive limits. For abelian C ∗algebras it coincides with covering ..."
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Cited by 15 (11 self)
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We introduce the completely positive rank, a notion of covering dimension for nuclear C ∗algebras and analyze some of its properties. The completely positive rank behaves nicely with respect to direct sums, quotients, ideals and inductive limits. For abelian C ∗algebras it coincides with covering dimension of the spectrum and there are similar results for continuous trace algebras. As it turns out, a C ∗algebra is zerodimensional precisely if it is AF. We consider various examples, particularly of onedimensional C ∗algebras, like the irrational rotation algebras, the BunceDeddens algebras or Blackadar’s simple unital projectionless C ∗algebra. Finally, we compare the completely positive rank to other concepts of noncommutative covering dimension, such as stable or real rank. The theory of C ∗algebras is often regarded as noncommutative topology, mainly
Amalgamated free products of C∗bundles
, 2007
"... Given two unital continuous C∗bundles A and B over the same compact Hausdorff base space X, we study the continuity properties of their different amalgamated free products over C(X). ..."
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Cited by 7 (5 self)
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Given two unital continuous C∗bundles A and B over the same compact Hausdorff base space X, we study the continuity properties of their different amalgamated free products over C(X).
Structure of graph C*algebras and their generalizations, Chapter in the book “Graph Algebras: Bridging the gap between analysis and algebra”, Eds. Gonzalo Aranda Pino, Francesc Perera Domènech, and Mercedes Siles Molina, Servicio de Publicaciones de la U
 Department of Mathematics, Keio University
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REAL RANK ZERO AND TRACIAL STATES OF C ∗ALGEBRAS ASSOCIATED TO GRAPHS
, 2002
"... Abstract. If G is a graph and C ∗ (G) is its associated C ∗algebra, then we show that the states on K0(C ∗ (G)) can be identified with T(G), the graph traces on G of norm 1. With this identification the standard map r C ∗ (G) : T(C ∗ (G))→S(K0(C ∗ (G))) from tracial states on C ∗ (G) to states on K ..."
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Cited by 3 (1 self)
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Abstract. If G is a graph and C ∗ (G) is its associated C ∗algebra, then we show that the states on K0(C ∗ (G)) can be identified with T(G), the graph traces on G of norm 1. With this identification the standard map r C ∗ (G) : T(C ∗ (G))→S(K0(C ∗ (G))) from tracial states on C ∗ (G) to states on K0(C ∗ (G)) becomes a map rG: T(C ∗ (G))→T(G). We prove that if G satisfies Condition (K), then the map rG is an affine homeomorphism. We also examine situations in which we can identify the extreme points of T(G). 1.
TRACIAL INVARIANTS, CLASSIFICATION AND II1 FACTOR REPRESENTATIONS OF POPA ALGEBRAS
, 2002
"... Abstract. Using various finite dimensional approximation properties, four convex subsets of the tracial space of a unital C ∗algebra are defined. One subset is characterized by Connes’ hypertrace condition. Another is characterized by hyperfiniteness of GNS representations. The other two sets are m ..."
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Cited by 3 (1 self)
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Abstract. Using various finite dimensional approximation properties, four convex subsets of the tracial space of a unital C ∗algebra are defined. One subset is characterized by Connes’ hypertrace condition. Another is characterized by hyperfiniteness of GNS representations. The other two sets are more mysterious but are shown to be intimately related to Elliott’s classification program. Applications of these tracial invariants include: 1. An analogue of Szegö’s Limit Theorem for arbitrary self adjoint operators. 2. A McDuff factor embeds into R ω if and only if it contains a weakly dense operator system which is injective. 3. There exists a simple, quasidiagonal, real rank zero C ∗algebra with nonhyperfinite II1 factor representations and which is not tracially AF. This answers negatively questions of Sorin Popa and, respectively, Huaxin Lin. 4. If A is any one of the standard examples of a stably finite, nonquasidiagonal C ∗algebra and B is a C ∗algebra with Lance’s WEP and at least one tracial state then there is no unital ∗homomorphism A → B. In particular, many stably finite, exact C ∗algebras
Homogeneity of the pure state space for the separable nuclear C ∗algebras
, 2001
"... We prove that the pure state space is homogeneous under the action of the group of asymptotically inner automorphisms for all the separable simple nuclear C ∗algebras. If simplicity is not assumed for the C ∗algebras, the set of pure states whose GNS representations are faithful is homogeneous for ..."
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Cited by 1 (0 self)
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We prove that the pure state space is homogeneous under the action of the group of asymptotically inner automorphisms for all the separable simple nuclear C ∗algebras. If simplicity is not assumed for the C ∗algebras, the set of pure states whose GNS representations are faithful is homogeneous for the above action. 1
On Quasidiagonal C*algebras
, 2000
"... We give a detailed survey of the theory of quasidiagonal C∗algebras. The main structural results are presented and various functorial questions around quasidiagonality are discussed. In particular we look at what is currently known (and not known) about tensor products, quotients, extensions, free ..."
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We give a detailed survey of the theory of quasidiagonal C∗algebras. The main structural results are presented and various functorial questions around quasidiagonality are discussed. In particular we look at what is currently known (and not known) about tensor products, quotients, extensions, free products, etc. of quasidiagonal C ∗algebras. We also point out how quasidiagonality is connected to some important open problems.