Marc A. Rieffel

Deformation quantization for actions of

Oscillatory integrals The deformed product Function algebras The algebra of bounded operators Functoriality for the operator norm Norms of deformed deformations Smooth vectors, and exactness Continuous fields Strict deformation quantization Old examples The quantum Euclidean closed disk and quantum quadrant The algebraists quantum plane, and quantum groups References.

Deformation quantization of Heisenberg manifolds

ForM a smooth manifold equipped with a Poisson bracket, we formulate aC*-algebra framework for deformation quantization, including the possibility of invariance under a Lie group of diffeomorphisms preserving the Poisson bracket. We then show that the much-studied non-commutative tori give examples of such deformation quantizations, invariant under the usual action of ordinary tori. Going beyond this, the main results of the paper provide a construction of invariant deformation quantizations for those Poisson brackets on Heisenberg manifolds which are invariant under the action of the Heisenberg Lie group, and for various generalizations suggested by this class of examples. Interesting examples are obtained of simpleC*-algebras on which the Heisenberg group acts ergodically.

Continuous Fields of C*-Algebras Coming from Group Cocycles and Actions

Recently I have been attempting to formulate a suitable C*-algebraic framework for the subject of deformation quantization [-3, 19]. Continuous fields of C*-algebras provide one of the key elements for this framework. The main examples of deformation quantizations which I have constructed up to now in this C*-algebra framework come from letting either cocycles on groups, or actions of groups, vary. It has thus become necessary to show that one obtains in this way fields of C*-aigebras that are indeed continuous. Since this material is of a general nature, and can be useful in other situations [4-6, 11-13] it has seemed appropriate to give a separate exposition of it, in the present article. Section 1 of this article contains a review of the published results on continuous fields which we will need, as well as a discussion of the fact that the approach which we will take involves treating upper and lower semi-continuity separately. In Sect. 2 we discuss the continuity of fields of C*-algebras which arise from varying cocycles on groups, while in Sect. 3 we do the same for actions which vary.

Integrable and proper actions on C*-algebras, and square-integrable representations of groups*

Abstract We propose a definition of what should be meant by a proper action of a locally compact group on a C *-algebra. We show that when the C *-algebra is commutative this definition exactly captures the usual notion of a proper action on a locally compact space. We then discuss how one might define a generalized fixed-point algebra . The goal is to show that the generalized fixed-point algebra is strongly Morita equivalent to an ideal in the crossed product algebra, as happens in the commutative case. We show that one candidate gives the desired algebra when the C *-algebra is commutative. But very recently Exel has shown that this candidate is too big in general. Finally, we consider in detail the application of these ideas to actions of a locally compact group on the algebra of compact operators (necessarily coming from unitary representations), and show that this gives an attractive view of the subject of square-integrable representations.

Proper Actions of Groups on C*-Algebras

Recently I have been attempting to formulate a suitable C*-algebraic framework for the subject of deformation quantization of Poisson manifolds [1,13]. Some of the main examples which I have constructed within this framework [27] involve “proper” actions of groups on C*-algebras, where “proper” actions are to be defined as a generalization of proper actions of groups on locally compact spaces. Much of the material on proper actions which I have developed for this purpose is of a general nature which may be useful in other situations, as it has seemed appropriate to give a separate exposition of it, in the present article.

Non-compact quantum groups associated with Abelian subgroups

AbstractLetG be a Lie group. For any Abelian subalgebra

Leibniz seminorms and best approximation from C * -subalgebras

\mathfrak{h}

Adjoint Functors and Derived Functors with an Application to the Cohomology of Semigroups

of the Lie algebra g ofG, and any