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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 ΔitMΔ−it = M and JMJ = M′, where M′ denotes 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.
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