Krieger factors isomorphic to their tensor square and pure point spectrum flows
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.
crossed product of an abelian von Neumann algebra by an ergodic automorphism
\(L^{\infty }\)-spectrum of the flow of weights of a factor
scientific article; zbMATH DE number 3940007
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