Krieger factors isomorphic to their tensor square and pure point spectrum flows
type: Item
instance of: scholarly article
title: Krieger factors isomorphic to their tensor square and pure point spectrum flows
zbMATH Open document ID: 0586.46048
DOI: 10.1016/0022-1236(85)90075-8
author: Georges Skandalis
author: Thierry Giordano
published in: Journal of Functional Analysis
publication date: 1985-01-01T00:00:00Z
review text: Infinite Krieger factors M (i.e. crossed product of an abelian von Neumann algebra by an ergodic automorphism) are completely classified by an ergodic flow, their flow of weights \(F_ M\). [\textit{W. Krieger}, Math. Ann. 223, 19-70 (1976; Zbl 0332.46045)]. If \(F_ M\) has (a finite invariant measure and) pure point spectrum, \textit{T. Hamachi, Y. Oka} and \textit{M. Osikawa} [Publ. Res. Inst. Math. Sci., Kyoto Univ. 11, 31-50 (1975; Zbl 0316.28007)] showed that M is isomorphic to \(M\otimes M\). The main result of this paper is a converse of this statement: if a factor M whose Sakai flip is approximately inner satisfies \(M=M\otimes M\), then its flow of weights (is finite measure preserving and) has pure point spectrum. (The Sakai flip \(\sigma\) of M is the automorphism such that \(\sigma (x\otimes y)=y\otimes x\), for all x,y\(\in M).\) The \(L^{\infty}\)-spectrum of the flow of weights of a factor is Connes' invariant T. We prove that for every ITPFI factor (infinite tensor product of factors of type I), there exists an infinite tensor product of factors of type \(I_ 2\) \((ITPFI_ 2\) factor) with the same Connes' invariant T. Examples of \(ITPFI_ 2\) factors related to the Krieger hierarchy (cf. the above reference) are also given.
Mathematics Subject Classification ID: 46L35
Mathematics Subject Classification ID: 46L55
zbMATH DE Number: 3940007
zbMATH Keywords: pure point spectrum flow
zbMATH Keywords: Infinite Krieger factors
zbMATH Keywords: crossed product of an abelian von Neumann algebra by an ergodic automorphism
zbMATH Keywords: ergodic flow
zbMATH Keywords: Sakai flip
zbMATH Keywords: \(L^{\infty }\)-spectrum of the flow of weights of a factor
zbMATH Keywords: ITPFI factor
zbMATH Keywords: infinite tensor product of factors
zbMATH Keywords: Connes' invariant
zbMATH Keywords: Krieger hierarchy
MaRDI profile type: Publication
full work available at URL: 0022-1236(85)90075-8
OpenAlex ID: W2056487814
cites work: A classification of factors
cites work: Q5660116
cites work: Une classification des facteurs de type ${\rm III}$
cites work: On the hierarchy of W. Krieger
cites work: Classification of injective factors. Cases \(\mathrm{II}_1\), \(\mathrm{II}_\infty\), \(\mathrm{III}_\lambda\), \(\lambda\neq 1\)
cites work: The flow of weights on factors of type III
cites work: A Construction of Approximately Finite-Dimensional Non-ITPFI Factors
cites work: Approximately transitive flows and ITPFI factors
cites work: Q3955608
cites work: Orbit structure and countable sections for actions of continuous groups
cites work: Q5615632
cites work: On infinite tensor products of factors of type I2
cites work: Q3881310
cites work: The normalizer group of an ergodic automorphism of type III and the commutant of an ergodic flow
cites work: Flows associated with ergodic non-singular transformation groups
cites work: Q3926822
cites work: A remark on characters of locally compact Abelian groups
cites work: On ergodic flows and the isomorphism of factors
cites work: Flow under a function and discrete decomposition of properly infinite \(W^*\)-algebras
cites work: Inverses, logarithms, and idempotents in M(G)
cites work: The Classification of Factors is not Smooth
cites work: Q3310409
cites work: Q4180808
description: scientific article; zbMATH DE number 3940007