Takeshi Katsura

A class of C*-algebras generalizing both graph algebras and homeomorphism C*-algebras I, fundamental results

We introduce a new class of C*-algebras, which is a generalization of both graph algebras and homeomorphism C*-algebras. This class is very large and also very tractable. We prove the so-called gauge-invariant unique; ness theorem and the Cuntz-Krieger uniqueness theorem, and compute the K-groups of our algebras.

A class of

We investigate the ideal structures of the C^*-algebras arising from topological graphs. We give the complete description of ideals of such C^*-algebras which are invariant under the so-called gauge action, and give the condition on topological graphs so that all ideals are invariant under the gauge action. We get conditions for our C^*-algebras to be simple, prime or primitive. We completely determine the prime ideals, and show that most of them are primitive. Finally, we construct a discrete graph such that the associated C^*-algebra is prime but not primitive.

{C^*}

We show that the method to construct C*-algebras from topological graphs, introduced in our previous paper, generalizes many known constructions. We give many ways to make new topological graphs from old ones, and study the relation of C*-algebras constructed from them. We also give a characterization of our C*-algebras in terms of their representation theory.

-algebras generalizing both graph algebras and homeomorphism

Abstract We prove that every action of a finite group all of whose Sylow subgroups are cyclic on the K-theory of a Kirchberg algebra can be lifted to an action on the Kirchberg algebra. The proof uses a construction of Kirchberg algebras generalizing the one of Cuntz-Krieger algebras, and a result on modules over finite groups. As a corollary, every automorphism of the K-theory of a Kirchberg algebra can be lifted to an automorphism of the Kirchberg algebra with same order.

{C^*}

We completely determine the ideal structures of the crossed products of Cuntz algebras by quasi-free actions of abelian groups and give another proof of A. Kishimotos result on the simplicity of such crossed products. We also give a necessary and sufficient condition that our algebras become primitive, and compute the Connes spectra and K-groups of our algebras.

-algebras III, ideal structures

Abstract We prove that the classes of graph algebras, Exel-Laca algebras, and ultragraph algebras coincide up to Morita equivalence. This result answers the long-standing open question of whether every Exel-Laca algebra is Morita equivalent to a graph algebra. Given an ultragraph we construct a directed graph E such that is isomorphic to a full corner of C*(E). As applications, we characterize real rank zero for ultragraph algebras and describe quotients of ultragraph algebras by gauge-invariant ideals.

A CLASS OF C*-ALGEBRAS GENERALIZING BOTH GRAPH ALGEBRAS AND HOMEOMORPHISM C*-ALGEBRAS II, EXAMPLES

There are many classes of nonsimple graph C*-algebras that are classified by the six-term exact sequence in K-theory. In this paper we consider the range of this invariant and determine which cyclic six-term exact sequences can be obtained by various classes of graph C*-algebras. To accomplish this, we establish a general method that allows us to form a graph with a given six-term exact sequence of K-groups by splicing together smaller graphs whose C*-algebras realize portions of the six-term exact sequence. As rather immediate consequences, we obtain the first permanence results for extensions of graph C*-algebras. We are hopeful that the results and methods presented here will also prove useful in more general cases, such as situations where the C*-algebras under investigations have more than one ideal and where there are currently no relevant classification theories available.

A construction of actions on Kirchberg algebras which induce given actions on their K-groups

Abstract We investigate crossed products of Cuntz algebras by quasi-free actions of abelian groups. We prove that our algebras are AF-embeddable when actions satisfy a certain condition. We also give a necessary and sufficient condition that our algebras become simple and purely infinite, and consequently our algebras are either purely infinite or AF-embeddable when they are simple.

The Ideal Structures of Crossed Products of Cuntz Algebras by Quasi-Free Actions of Abelian Groups

Abstract We augment Restorffs classification of purely infinite Cuntz–Krieger algebras by describing the range of his invariant on purely infinite Cuntz–Krieger algebras. We also describe its range on purely infinite graph C ⁎ -algebras with finitely many ideals, and provide ‘unital’ range results for purely infinite Cuntz–Krieger algebras and unital purely infinite graph C ⁎ -algebras.

GRAPH ALGEBRAS, EXEL-LACA ALGEBRAS, AND ULTRAGRAPH ALGEBRAS COINCIDE UP TO MORITA EQUIVALENCE

We give two pathological phenomena for non-separable AF-algebras which do not occur for separable AF-algebras. One is that non-separable AF-algebras are not determined by their Bratteli diagrams, and the other is that there exists a non-separable AF-algebra which is prime but not primitive.