Functional Analysis

By means of techniques from the Morita equivalence theory, we get finitely generated and projective modules over the quantum Heisenberg manifolds. This enables us to get some information about the range of the trace of these algebras, at the level of the K_{0}-group, in terms of the Poisson bracket in whose direction the manifolds are deformed.

Aim of this paper is to develop a new technique, based on the Baire category theorem, in order to establish the closure of reachable sets and the existence of optimal trajectories for control systems, without the usual convexity assumptions. The bang-bang property is proved for a new class of ``concave" multifunctions, characterized by the existence of suitable linear selections. The proofs rely on Lyapunov's theorem in connection with a Baire category argument.

Starting from Kirchberg's theorems announced in 1994, namely O_2 tensor A is isomorphic to O_2 for separable unital nuclear simple A and O_infinity tensor A is isomorphic to A if in addition A is purely infinite, we prove that KK-equivalence implies isomorphism for nonunital separable nuclear purely infinite simple C*-algebras. It follows that if A and B are unital separable nuclear purely infinite simple C*-algebras which satisfy the Universal Coefficient Theorem, and if there is a graded isomorphism from K_* (A) to K_* (B) which preserves the class of the identity, then A is isomorphic to B. Our main technical results are, we believe, of independent interest. We say that two asymptotic morphisms t ---> \phi_t and t ---> \psi_t from A to B are asymptotically unitarily equivalent if there exists a continuous unitary path t ---> u_t in the unitization B^+ such that || u_t \phi_t (a) u_t^* - \psi_t (a) || ---> 0 for all a in A. Let A be separable, nuclear, unital, and simple, and let D be unital. We prove that any asymptotic morphism from A to K tensor O_infinity tensor D is asymptotically unitarily equivalent to a homomorphism, and two homotopic homomorphisms from A to K tensor O_infinity tensor D are asymptotically unitarily equivalent.

This paper contains a complete description of classes of the unitary equivalence of the admissible representations of infinite-dimensional classic matrix groups paper.

We show that if E is a Frechet G\rtimes S(M)-module, for which the canonical map from the projective completion G\rtimes S(M) {\widehat \otimes} E to E is surjective, then every element of E can be written as a finite sum of elements of the form ae where e\in E and a is an element of the smooth crossed product G\rtimes S(M). We require that the Schwartz functions S(M) vanish rapidly with repsect to a continuous, proper map \s : M ---> [0, \infty).

Given C*-algebras A and B and an imprimitivity A-B-bimodule X, we construct an explicit isomorphism X_* : K_i(A) --> K_i(B) where K_i denote the complex K-theory functors for i=0, 1. Our techniques do not require separability nor existence of countable approximate identities. We thus extend, to general C*-algebras, the result of Brown, Green and Rieffel according to which strongly Morita equivalent C*-algebras have isomorphic K-groups. The method employed includes a study of Fredholm operators on Hilbert modules.

Let M be a factor with separable predual and G a compact group of automorphisms of M whose action is minimal, i.e. M G ′ ∩M=C , where M G denotes the G -fixed point subalgebra. Then every intemediate von Neumann algebra M G ⊂N⊂M has the form N= M H for some closed subgroup H of G . An extension of this result to the case of actions of compact Kac algebras on factors is also presented. No assumptions are made on the existence of a normal conditional expectation onto N .

Complex analyticity is generalized to hypercomplex functions, quaternion or octonion, in such a manner that it includes the standard complex definition and does not reduce analytic functions to a trivial class. A brief comparison with other definitions is presented.

We give an example of a dense, simple, unital Banach subalgebra A of the irrational rotation C*-algebra B , such that A is not a spectral subalgebra of B . This answers a question posed in T.W. Palmer's paper [1].

A brief introduction into bimodules of I I 1 -factors is presented. Furthermore a version of the following result due to M. Pimsner and S. Popa is derived: Let N= M −1 ⊂M= M 0 ⊂ M 1 ⊂ M 2 ⊂… denote the Jones tower of a I I 1 -factor N⊂M with finite index. Then the factor obtained by the basic construction from the pair N⊂ M n−1 is equal to M 2n−1 .