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Standard weights on algebras of unbounded operators. (English) Zbl 0953.47048

For topological \( O^* \)-algebras \(\mathbb M\) of unbounded Hilbert-space operators [see K. Schmüdgen, “Unbounded operator algebras and representation theory”, Berlin (1990; Zbl 0697.47048)], known earlier as \(Op^* \)-algebras [cf., e.g., G. Lassner, Rep. Math. Phys. 3, 279-293 (1972; Zbl 0252.46087)], the paper continues analysis of modular theory related to weights and quasi-weights \( \varphi \) introduced by A. Inoue and H. Ogi [in J. Math. Soc. Japan 50, No. 1, 227-252 (1998; Zbl 0904.47037)]. A useful GNS-representation \( \pi_\varphi \) of \(\mathbb M\), and accompanying vector representation \( \lambda_\varphi \) of the left ideal \( {\mathcal N}_\varphi=\{X\in \mathbb M:\varphi((AX)^+AX)<\infty, \forall A\in \mathbb M\} \), arise in the case \( (\#) \) of faithful \(\sigma\)-weakly continuous semifinite (quasi-) weight \( \varphi \), whenever the weak commutant \( \pi_\varphi(\mathbb M)'_W \) is a von Neumann algebra on the representation space \( {\mathcal H}_\varphi \). The notions of quasi-standardness and standardness of generalized vectors for the \( O^* \)-algebra \( \pi_\varphi(\mathbb M) \) defined by A. Inoue [in J. Math. Soc. Japan 47, No. 2, 329-347 (1995; Zbl 0884.47023)] are then carried over to \( \varphi \). n particular a standard (quasi-) weight \( \varphi \) is shown to satisfy KMS-condition with respect to the natural modular automorphism group \( \sigma_t^\varphi \) of \(\mathbb M \). A generalized Connes cocycle theorem for weights \( \varphi \) and \( \psi \) is established. It is close to the classical \( W^* \)-algebra result if \( \pi_\varphi \) and \( \pi_\psi \) are unitarily equivalent and \( \pi_\varphi(\mathbb M) \) is a generalized von Neumann algebra. The latter occurs if \(\mathbb M\) is a generalized von Neumann algebra with a strongly dense bounded part and \( \varphi \) meets the above conditions \( (\#) \).

MSC:

47L60 Algebras of unbounded operators; partial algebras of operators
46L10 General theory of von Neumann algebras
46K10 Representations of topological algebras with involution
46L55 Noncommutative dynamical systems
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