Author
Richard H. Herman
Bio: Richard H. Herman is an academic researcher from University of California, Los Angeles. The author has contributed to research in topics: Automorphism & Inner automorphism. The author has an hindex of 2, co-authored 2 publications receiving 62 citations.
Papers
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TL;DR: In this article, it was shown that the discrete spectrum of an abelian unitary automorphism group acting as automorphisms of a von Neumann algebra is characterized by elements in the CAR algebra.
Abstract: Suppose that a group of automorphisms of a von Neumann algebraM, fixes the center elementwise We show that if this group commutes with the modular (KMS) automorphism group associated with a normal faithful state onM, then this state is left invariant by the group of automorphisms As a result we obtain a “noncommutative” ergodic theorem The discrete spectrum of an abelian unitary group acting as automorphisms ofM is completely characterized by elements inM We discuss the KMS condition on the CAR algebra with respect to quasi-free automorphisms and gauge invariant generalized free states We also obtain a necessary and sufficient condition for the CAR algebra and a quasi-free automorphism group to be η-abelian
65 citations
TL;DR: In this article, it was shown that strong (pointwise) convergence of modular automorphism groups to a one parameter family of maps implies weak convergence of the respective states in the factor case.
Abstract: It is shown, under a necessary condition, that strong (pointwise) convergence of modular automorphism groups to a one parameter family of maps implies weak convergence of the respective states in the factor case. Moreover the limiting one parameter family of maps is the modular automorphism group for the limiting state. In the type I case weak convergence of the automorphism groups suffices. Norm convergence of the states is obtained in some cases.
2 citations
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TL;DR: In this article, it was shown that a von Neumann algebra M is invariant under modular automorphism group σtϑ associated with ϑ if and only if there exists a σ-weakly continuous faithful projection ϵ of norm one from M onto N such that ϑ(x) = ϑ ∘ ϵ(x), for every xϵMϑ.
492 citations
TL;DR: In this paper, the authors give an essentially self-contained account of structural properties of quantum open Markovian systems and discuss a general form of quantum detailed balance and its relation to thermal relaxation and to microreversibility.
412 citations
TL;DR: The structure of a v o n Neumann Neumann algebra of type I I I... ε I I ε, ε, ε is invariant to S ( ~ ) and T ( ~) of A. Connes.
Abstract: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Preliminary 251 Construction of crossed products . . . . . . . . . . . . . . . . . . . . . . . 253 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 Dual weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Bi-dual weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 Subgroups and subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . 284 The structure of a v o n Neumann algebra of type I I I . . . . . . . . . . . . . . 286 Algebraic invariants S ( ~ ) and T ( ~ ) of A. Connes . . . . . . . . . . . . . . . 294 Induced action and crossed products . . . . . . . . . . . . . . . . . . . . . 297 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
337 citations
Book•
01 Jan 1970
TL;DR: In this article, the commutation theorem for modular Hilbert algebras has been proved for generalized Hilbert algebra and the modular automorphism group has been shown to have semi-finiteness.
Abstract: Preliminaries.- Modular Hilbert algebras.- Generalized Hilbert algebras.- The commutation theorem for modular Hilbert algebras.- Self-adjoint subalgebras of generalized Hilbert algebras.- The spectral algebra.- The modular operator ?.- The resolvent of the modular operator ?.- The one-parameter automorphisms defined by the modular operator ?.- Formulation of the modular Hilbert algebra.- Tensor product and direct sum of modular Hilbert algebras.- The standard representation of von Neumann algebras.- The modular automorphism group and the Kubo-Martin-Schwinger boundary condition.- Semi-finiteness and the modular automorphism group.
252 citations
TL;DR: In this article, it was proved that injective factors of type III,t, 2. 1 on a separable Hilbert space are completely classified by their smooth flow of weights.
Abstract: In Connes' fundamental work "Classification of injective factors" [7], it is proved that injective factors of type III,t, 2 . 1 on a separable Hilbert space are completely classified by their "smooth flow of weights". Since the flow of weights of factors of type III1 is trivial, one would expect that there is only one isomorphism class of injective factors of type IIIt. During the years 1976-78, Connes spent much effort to prove that there is only one injective factor of type III1, and found a number of conditions for an injective factor of type III1 to be isomorphic to the Araki-Woods' factor R~o. One of these conditions is the following: Let q0 be a normal faithful state on a yon Neumann algebra M, and let the bicentralizer of q0 be the set B. of operators a in M for which
220 citations