Pimsner introduced the C*-algebra O X generated by a Hilbert bimodule X over a C*-algebra A. We look for additional conditions that X should satisfy in order to study the simplicity and, more generally, the ideal structure of O X when X is finite projective. We introduce two conditions,``(I)-freeness'' and``(II )-freeness,'' stronger than the former, in analogy with J. the case of the bimodules associated with an inclusion of simple C*-algebras with finite index, real or pseudoreal bimodules with finite intrinsic dimension, and the case of``Cuntz Krieger bimodules.'' If X satisfies this condition the C*-algebra O X does not depend on the choice of the generators when A is faithfully represented. As a consequence, if X is (I )-free and A is X-simple, then O X is simple. In the case of Cuntz Krieger algebras O A , X-simplicity corresponds to the irreducibility of the matrix A. If A is simple and p.i. then O X is p.i.; if A is nonnuclear then O X is nonnuclear. Thus we provide many examples of (purely) infinite nonnuclear simple C*-algebras. Furthermore if X is (II )-free, we determine the ideal structure of O X . 1998 Academic Press article no. FU983306 295

The spectral functor of an ergodic action of a compact quantum group G on a unital C * -algebra is quasitensor, in the sense that the tensor product of two spectral subspaces is isometrically contained in the spectral subspace of the tensor product representation, and the inclusion maps satisfy natural properties. We show that any quasitensor * -functor from Rep(G) to the category of Hilbert spaces is the spectral functor of an ergodic action of G on a unital C * -algebra.As an application, we associate an ergodic G-action on a unital C *algebra to an inclusion of Rep(G) into an abstract tensor C * -category T.If the inclusion arises from a quantum subgroup K of G, the associated G-system is just the quotient space K\G. If G is a group and T has permutation symmetry, the associated G-system is commutative, and therefore isomorphic to the classical quotient space by a subgroup of G.If a tensor C * -category has a Hecke symmetry making an object ρ of dimension d and µ-determinant 1 then there is an ergodic action of SµU (d) on a unital C * -algebra having the (ι, ρ r ) as its spectral subspaces. The special case of SµU (2) is discussed.

We introduce the notion of finite right (or left) numerical index on a C Ã -bimodule A X B with a bi-Hilbertian structure, based on a Pimsner-Popa-type inequality. The right index of X can be constructed in the centre of the enveloping von Neumann algebra of A . The bimodule X is called of finite right index if the right index lies in the multiplier algebra of A: In this case the Jones basic construction enjoys nice properties. The C Ã -algebra of bimodule mappings with a right adjoint is a continuous field of finite dimensional C Ã -algebras over a compact Hausdorff space, whose fiber dimensions are bounded above by the index. If A is unital, the right index belongs to A if and only if X is finitely generated as a right module. A finite index bimodule is a bi-Hilbertian C Ã -bimodule which is at the same time of finite right and left index.Bi-Hilbertian, finite index C Ã -bimodules, when regarded as objects of the tensor 2-C Ã -category of right Hilbertian C Ã -bimodules, are precisely those objects with a conjugate in the same category, in the sense of Longo and Roberts. r

Abstract. To any periodic and full C * -dynamical system (A, α, R), an invertible operator s acting on the Banach space of trace functionals of the fixed point algebra is canonically associated. KMS states correspond to positive eigenvectors of s. A Perron-Frobenius type theorem asserts the existence of KMS states at inverse temperatures equal the logarithms of the inner and outer spectral radii of s (extremal KMS states). Examples arising from subshifts in symbolic dynamics, self-similar sets in fractal geometry and noncommutative metric spaces are discussed.Certain subshifts are naturally associated to the system, and criteria for the equality of their topological entropy and inverse temperatures of extremal KMS states are given.Unital completely positive maps σ {x j } implemented by partitions of unity {x j } of grade 1 are considered, resembling the 'canonical endomorphism' of the Cuntz algebras. The relationship between the Voiculescu topological entropy of σ {x j } and the topological entropy of the associated subshift is studied. Examples where the equality holds are discussed among Matsumoto algebras associated to non finite type subshifts. In the general case ht(σ {x j } ) is bounded by the sum of the entropy of the subshift and a suitable entropic quantity of the homogeneous subalgebra. Both summands are necessary.The measure-theoretic entropy of σ {x j } , in the sense of Connes-NarnhoferThirring, is compared to the classical measure-theoretic entropy of the subshift.A noncommutative analogue of the classical variational principle for the entropy is obtained for the 'canonical endomorphism' of certain Matsumoto algebras. More generally, a necessary condition is discussed. In the case of CuntzKrieger algebras an explicit construction of the state with maximal entropy from the unique KMS state is done.

We consider the class of``localized endomorphisms'' of the Cuntz algebras and we make some computations on the index of the associated endomorphisms of type III * factors.