No abstract

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.

Abstract. 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 uniqueness theorem and the Cuntz-Krieger uniqueness theorem, and compute the K-groups of our algebras. IntroductionThe purpose of this serial work is an introduction of a new class of C * -algebras which contains graph algebras and homeomorphism C * -algebras. Our class is very large so that it contains every AF-algebra [Ka2] and every Kirchberg algebra satisfying the UCT [Ka4] as well as many simple stably projectionless C * -algebras. At the same time, our class can be well studied by using similar techniques developed in the analysis of graph algebras and homeomorphism C * -algebras. Since J. Cuntz and W. Krieger introduced a class of C * -algebras arising from finite matrices with entries {0, 1} in [CK], there have been many generalizations of Cuntz-Krieger algebras, for example, Exel-Laca algebras [EL], graph algebras [KPRR, KPR, FLR] and Cuntz-Pimsner algebras [P]. Among others, investigation of graph algebras has rapidly progressed these days (see, for example, [BPRS, BHRS, HS, DT1]), and many structures of graph algebras have been characterized in terms of graphs. As some authors pointed out, it is time to extend the techniques and results on graph algebras to more general C * -algebras. Our work is one of such attempts. The investigation of homeomorphism C * -algebras has also been developed mainly by J. Tomiyama [T1, T2, T3, T4]. These two lines of research have several similar aspects in common, and our aim in this series of work is to combine and unify these studies in the two active fields.In this paper, we associate a C * -algebra with a quadruple E = (E 0 , E 1 , d, r) where E 0 and E 1 are locally compact spaces, d : E 1 → E 0 is a local homeomorphism, and r :is called a topological graph. Note that when E 0 is a discrete set, this quadruple is an ordinary (directed) graph and the C * -algebra constructed here is a graph algebra of it (or its opposite graph Ka3]).In the first paper of our serial work, we give a definition of our algebras and prove fundamental results on them. We first construct C * -correspondences from topological graphs. This is done in Section 1 in a slightly more general form. Then in Section 2, we associate C * -algebras with C * -correspondences constructed from topological graphs, in a similar way to Cuntz-Pimsner algebras [P]. We, however, point out two distinctions between our approach and the one for Cuntz-Pimsner algebras (see also the end of Section 3 in this paper). The first point is that left actions of our C * -correspondences may not be injective. This is not allowed in [P] because Cuntz-Pimsner algebras of C * -correspondences with non-injective left actions often become zero (see [P, Remark 1.2 (1)]). Note that our algebras are obtained as relative Cuntz-Pimsner algebras introduced in [MS]. The other point is that we can examine our algebr...

We study C Ã -algebras arising from C Ã -correspondences, which were introduced by the author. We prove the gauge-invariant uniqueness theorem, and obtain conditions for our C Ã -algebras to be nuclear, exact, or satisfy the Universal Coefficient Theorem. We also obtain a 6-term exact sequence of K-groups involving the K-groups of our C Ã -algebras. r 2004 Elsevier Inc. All rights reserved. MSC: Primary 46L05; 46L55