tailieunhanh - Báo cáo toán học: "The spectral category and the Connes invariant $\Gamma$ "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Các thể loại quang phổ và Connes bất biến $ Gamma \ $. | Copyright by INCREST 1985 J. OPERATOR THEORY 14 1985 129-146 THE SPECTRAL CATEGORY AND THE CONNES INVARIANT r p. GHEZ R. LIMA J. E. ROBERTS INTRODUCTION In this paper we describe a new approach to the harmonic analysis of the action of a locally compact group G on a von Neumann algebra M. In particular one would like to understand how the action of G on the linear space M is related to the algebraic structure of M. In the context of von Neumann algebras the natural step is to compare the action of G on M with the continuous unitary representations of G and to relate the product and adjoint in M to the tensor product and conjugation of such representations. When G is abelian the theory of spectral subspaces is a useful tool for analyzing the action. This theory was first formalized by Arveson 1 although most of the basic techniques had long been used in theoretical physics. For non-abelian groups no such formalism exists as yet although there is again a long practical tradition in theoretical physics of treating particularly the action of compact non-abelian groups in terms of multiplets of operators transforming according to a given irreducible representation Ơ of G. The corresponding spectral subspace M appears naturally as a subspace of H M rather than of M itself. Here H is the Hilbert space of the representation a and 77 its dual space. Spectral subspaces of this type were introduced in 16 in the case of a compact group and will be employed here in preference to the subspaces of M proposed by Evans and Sund 6 as they have the advantage of exhibiting explicitly the transformation law under G. For non-abelian groups the spectral subspaces on their own are not much use but must be seen as part of a wider structure termed the spectral category which includes the algebras of spherical functions of Landstad 8 . These algebras arise naturally in studying spectral subspaces because whereas the left support of Ma is in the fixed-point algebra its right support is a .