Vol. 69, No. 1, 1977

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A bounded operator approach to Tomita-Takesaki theory

Marc A. Rieffel and Alfons Van Daele

Vol. 69 (1977), No. 1, 187–221

Let M be a von Neumann algebra. The Tomita-Takesaki theory associates with each cyclic and separating vector for M a strongly continuous one-parameter group, Δit, of unitary operators and a conjugate-linear isometric involution, J, of the underlying Hilbert space, such that ΔitMΔit = M and JMJ = M, where Mdenotes the commutant of M.

The present paper has two purposes. In the first half of the paper we show that the operators Δit and J can, in fact, be associated with any fairly general real subspace of a complex Hilbert space, and that many of their properties, for example the characterization of Δit in terms of the K.M.S. condition, can be derived in this less complicated setting. In the second half of the paper we show, by using some of the ideas from the first half, that a simplified proof of the Tomita-Takesaki theory given recently by the second author can be reformulated entirely in terms of bounded operators, thus further simplifying it by, among other things, eliminating all considerations involving domains of unbounded operators.

Mathematical Subject Classification 2000
Primary: 46L10
Received: 13 October 1976
Published: 1 March 1977
Marc A. Rieffel
University of California, Berkeley
Berkeley CA
United States
Alfons Van Daele
Department of Mathematics
Katholieke Universiteit Leuven
3030 Leuven