One-Parameter Groups Arising from Real Subspaces of Self-Dual Hilbert W*-Moduli
✍ Scribed by Michael Frank
- Publisher
- John Wiley and Sons
- Year
- 1990
- Tongue
- English
- Weight
- 851 KB
- Volume
- 145
- Category
- Article
- ISSN
- 0025-584X
No coin nor oath required. For personal study only.
✦ Synopsis
The purpose of this paper is to generalize the results of M. A. RIEFPEL and A. VAN
DAELE
[8, 30 1, 2,3] for HILBEET W*-moduli over commutative W*-Algebras. Some special real sub-@paces of such HXLBERT W*-moduli and the related operators are investigated. Particularly, the relation is established between *weakly continuous unitary one-parameter groups of operators arising from them and the generalized K.M.S. condition. All key definitions are formulated without any commutativity supposition for the underlying W*-algebra. The interpretation of these results is given for sets of continuous sections of "self-dual" locally trivial HILBERT bundles over hyperstonian compact spaces. At the end of this paper some aspects of the general noncommutative case are discussed.
8 1
Math. Nechr. 145 (1990) We remark that, in general, the topology T? is weaker than the topology tl, and that they both are weaker than the norm topology. Throughout this paper we use the following notation. If X is a subset of the HILBEBT W*-module 2, [X]T denotes the set {h: 1 E R , , z E Xo} where Xo is the t,-completion of the set {z E X: 1 1 z 1 1 5 1). Theorem 1.2 [4, Th. 91. Let A be a W*-aQebra and 3e be a HILBEBT A-module. The following conditions are equivalent:
(ii) The unit ball of 3e is complete with rmpect to the toplogy tl, i.e. X = [XK. (iii) The unit ball of 3e is crnnplete with respect to the top-y t 2 . Corollary 1.3 [4, Cor. 111. If A is a W*+ebra and X is a self-dual -BERT Amodule the linear span of the range of the A -v a l d inner product on 3e becames both a. W*-subalgebra and an &-leal in A. Theorem 1.4 "7, Prop. 3.10.1. Let A be a W*-algebra and X be a self-dual HILBEBT A-module. Then, the set EndA (a) of all bounded A-linear operators on d is a W*-alqebra.
These facts make clear that in the caae of 3t' being a selfdual &BEET W*-module, the spectral theorem ([lo, Th. 1.11.3.1) ie valid for each self-adjoint element of EndA (X).
Moreover, there exists a polar decomposition for each element of EndA (a) in EndA (d).