Elements of Tomita-Takesaki theory for embeddable AW*-algebras

Abstract

In this paper a Tomita-Takesaki type theorem for embeddable AW*-algebras M1 possessing in one of their "normal" representations on a faithful self-dual Hilbert 3(/)-module * a cyclic-separating element x E f is proved. The proving techniques are derived mainly from a paper of M. A. Rieffel and A. van Daele with some changes and generalizations. The crucial role in the proof play some special real subspaces oX of f generated by the self-adjoint part of T1~ and by x E #f, and operators related to X. The results from their investigation are of more general interest. § 1. IntroductionIn this paper we prove a Tomita-Takesaki type theorem for embeddable AW*-algebras possessing in one of their "normal" representations a cyclic-separating element. For this aim we use the way of proving determined in a paper of M. A. Rieffel and A. van Daele ([16]) with some changes in the proving techniques.The organization of the present paper is as follows. The second paragraph is concerned with basic definitions and with facts related to the subject. Under the supposition that X' is a self-dual Hilbert WI-module over a certain commutative AW*-algebra I we consider some special real self-dual Hilbert ¶Wh-submoduli X of ' and some bounded operators arising from them in the third paragraph. In particular, a conjugate-91-linear involution J on X with respect to oX is defined. In paragraph four order continuous unitary one-parameter groups {/izt:t E R} are investigated being defined for each such real subspace o-of Xf on Xf. The relation of those groups to a generalized K.M.S. condition is established. (Similar results are contained in a preprint of H. Halpern ([7], private communication)). We remark that the structure of the paragraphs three and four was suggested by [16]. Paragraph five is concerned with the Tomita-Takesaki type theorem for embeddable AW*-algebras possessing in one of their "normal" representations a cyclic-separating element. A interpretation of the results of the former paragraphs in terms of locally trivial Hilbert bundles over stoneian compact spaces is given in paragraph six. § 2. PreliminariesIn this part of the present paper the restriction on 91 to be commutative is often superfluous. But it will be necessary in further discussions.