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. The 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-algebras [Ka2] and every Kirchberg algebras satisfying the UCT [Ka4] as well as many simple stably projectionless C ∗-algebras. At

We associate to each locally finite directed graph G two locally compact groupoids G and G(?). The unit space of G is the space of one--sided infinite paths in G, and G(?) is the reduction of G to the space of paths emanating from a distinguished vertex ?. We show that under certain conditions their C --algebras are Morita equivalent; the groupoid C --algebra C (G) is the Cuntz--Krieger algebra of an infinite f0; 1g matrix defined by G, and that the algebras C (G(?)) contain the C --algebras used by Doplicher and Roberts in their duality theory for compact groups. We then analyse the ideal structure of these groupoid C --algebras using the general theory of Renault, and calculate their K-theory. 1 Introduction Over the past fifteen years many C -algebras and classes of C -algebras have been given groupoid models. Here we consider locally finite directed graphs, which may have infinitely many vertices, but only finitely many edges in and out of each vertex. We associate ...

We regard a right Hilbert C ∗ –module X over a C ∗ –algebra A endowed with an isometric ∗ –homomorphism φ: A → LA(X) as an object XA of the C ∗ –category of right Hilbert A–modules. Following [11], we associate to it a C ∗ –algebra OXA containing X as a “Hilbert A–bimodule in OXA ”. If X is full and finite projective OXA is the C ∗ –algebra C ∗ (X) , the generalization of the Cuntz–Krieger algebras introduced by Pimsner [27]. More generally, C ∗ (X) is canonically embedded in OXA as the C ∗ –subalgebra generated by X. Conversely, if X is full OXA is canonically embedded in C ∗ (X) ∗ ∗. Moreover, regarding X as an object AXA of the C ∗ –category of Hilbert A–bimodules, we associate to it a C ∗ –subalgebra OAXA of OXA commuting with A, on which X induces a canonical endomorphism ρ. We discuss conditions under which A and OAXA are the relative commutant of each other and X is precisely the subspace of intertwiners in OXA between the identity and ρ on O AXA. We also discuss conditions which imply the simplicity of C ∗ (X) or of OXA; in particular, if X is finite projective and full, C ∗ (X) will be simple if A is X–simple and the “Connes spectrum ” of X is T. 1

Abstract. We consider the norm closure A of the algebra of all operators of order and class zero in Boutet de Monvel’s calculus on a manifold X with boundary ∂X. We first describe the image and the kernel of the continuous extension of the boundary principal symbol homomorphism to A. If X is connected and ∂X is not empty, we then show that the K-groups of A are topologically determined. In case the manifold, its boundary, and the cotangent space of its interior have torsion free K-theory, we get Ki(A/K) ≃ Ki(C(X))⊕K1−i(C0(T ∗ ˙ X)), i = 0,1, with K denoting the compact ideal, and T ∗ ˙ X denoting the cotangent bundle of the interior. Using Boutet de Monvel’s index theorem, we also prove that the above formula holds for i = 1 even without this torsion-free hypothesis. For the case of orientable, two-dimensional X, K0(A) ≃ Z 2g+m and K1(A) ≃ Z 2g+m−1, where g is the genus of X and m is the number of connected components of ∂X. We also obtain a composition sequence 0 ⊂ K ⊂ G ⊂ A, with A/G commutative and G/K isomorphic to the algebra of all continuous functions on the cosphere bundle of ∂X with values in compact operators on L 2 (R+). 1.

We construct the C∗-algebra C(Lq(p; m1,...,mn)) of continuous functions on the quantum lens space as the fixed point algebra for a suitable action of Zp on the algebra C(S 2n−1 q corresponding to the quantum odd dimensional sphere. We show that C(Lq(p; m1,...,mn)) is isomorphic to the graph algebra C∗ ( L (p;m1,...,mn) 2n−1. This allows us to determine the ideal structure and, at least in principle, calculate the K-groups of C(Lq(p; m1,...,mn)). Passing to the limit with natural imbeddings of the quantum lens spaces we construct the quantum infinite lens space, or the quantum Eilenberg-MacLane space of type (Zp, 1).

Abstract. We show that two C ∗-algebraic noncommutative tori are strongly Morita equivalent if and only if they have isomorphic ordered K0-groups and centers, extending N. C. Phillips’s result in the case that the algebras are simple. This is also generalized to the twisted group C ∗-algebras of arbitrary finitely generated abelian groups. 1.

Abstract. In this paper the concept of unbounded Fredholm operators on Hilbert C ∗-modules over an arbitrary C ∗-algebra is discussed and the Atkinson theorem is generalized for bounded and unbounded Feredholm operators on Hilbert C ∗-modules over C ∗-algebras of compact operators. In the framework of Hilbert C ∗-modules over C ∗-algebras of compact operators, the index of an unbounded Fredholm operator and the index of its bounded transform are the same. 1. Introduction. Hilbert C ∗-modules are an often used tool in operator theory and in operator algebras. The theory of Hilbert C ∗-modules is very interesting on its own. Interacting with the theory of operator algebras and including ideas from non-commutative geometry it progresses and produces results and new problems attracting attentions (see [4]).

. We introduce the notion of the crossed product A oX Zof a C - algebra A by a Hilbert C -bimodule X. It is shown that given a C -algebra B, which carries a semi-saturated action of the circle group (in the sense that B is generated by the spectral subspaces B 0 and B 1 ), then B is isomorphic to the crossed product B 0 oB1 Z. We then present our main result, in which we show that the crossed products A oX Zand B oY Zare strongly Morita equivalent to each other, provided that A and B are strongly Morita equivalent under an imprimitivity bimodule M satisfying X\Omega A M ' M\Omega B Y as A \Gamma B Hilbert C - bimodules. We also present a six-term exact sequence for K-groups of crossed products by Hilbert C -bimodules. 1. Introduction If A is a C -algebra and ff is an automorphism of A, let B be the crossed product C -algebra B = Ao ff Z([16]). It is well known that B carries a natural action of the circle group, called the dual action, such that the spectra...

Abstract. A generalization of Connes-Thom isomorphism is given for stable, homotopy invariant, and split exact functors on separable C ∗-algebras. As examples of these functors, we concentrate on asymptotic and local cyclic cohomology and the result is applied to improve some formulas in asymptotic and local cyclic cohomology of C ∗-algebras. As an other application, it is shown that these cyclic theories are rigid after Rieffel’s deformation quantizations.

Abstract. We show that two C ∗-algebraic noncommutative tori are strongly Morita equivalent if and only if they have isomorphic ordered K0-groups and centers, extending N. C. Phillips’s result in the case that the algebras are simple. This is also generalized to the twisted group C ∗-algebras of arbitrary finitely generated abelian groups. 1.