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Interpolation by type I factors and the flip automorphism. (English) Zbl 0535.46036
For two von Neumann algebras \({\mathfrak A}\subset {\mathfrak B}\), the relation between the existence of an intermediate type I factor \({\mathfrak E} ({\mathfrak A}\subset {\mathfrak E}\subset {\mathfrak B})\) and the implementability of the flip automorphism \(\sigma\) of \(\tilde {\mathfrak A}={\mathfrak A}\otimes {\mathfrak B} (\sigma(a\otimes b)=b\otimes a)\) by a unitary in \(\tilde {\mathfrak B}={\mathfrak B}\otimes {\mathfrak B}\) is investigated.
It is first shown (Theorem 1) that the following 3 conditions are equivalent:
(i) The map \(a\otimes b'\to ab'\) (\(a\in {\mathfrak A}\), b’\(\in {\mathfrak B}')\) extends to an isomorphism of \({\mathfrak A}\otimes {\mathfrak B}\) onto \({\mathfrak A}\cup {\mathfrak B}')''.\)
(ii) There exists a type I factor \({\mathfrak F}\) such that \({\mathfrak A}\otimes 1\subset {\mathfrak F}\subset {\mathfrak B}\otimes B({\mathfrak K}),\), \(\dim {\mathfrak K}=\infty.\)
(iii) For flip automorphism \(\sigma\) of \(\tilde {\mathfrak A}\) there exist projections \(e_ 1,e_ 2\in \tilde {\mathfrak A}'\cap \tilde {\mathfrak B}\) with central support 1 and a partial isometry \(v\in \tilde {\mathfrak B}\) such that \(v^*v=e_ 1, vv^*=e_ 2\) and \(\sigma(a)e_ 1=v^*av\) for all \(a\in \tilde {\mathfrak B}\). Such a \(\sigma\) is said to be quasi-\(\tilde{\mathfrak B}\)-inner (\(\tilde{\mathfrak B}\)-inner if \(e_ 1=e_ 2=1).\)
It is then shown (Theorem 2) under 6 different sufficient conditions that (iii) above implies the existence of an intermediate type I factor \({\mathfrak F}\). This includes a special case of \({\mathfrak A}={\mathfrak B}\) leading to a result of D. Sakai [Am. J. Math. 97, 1975, 889-896 (1976; Zbl 0321.46052)]. On the other hand it is shown by counterexamples that \(\tilde {\mathfrak B}\)-inner \(\sigma\) is quasi-\(\tilde{\mathfrak B}\)-inner but not vice versa and the existence of an intermediate type I factor implies \(\tilde {\mathfrak B}\)-innerness of \(\sigma\) but not vice versa.
The result is applied to a new proof of the existence of an intermediate type I factor for local algebras of a free scalar field [a result of D. Buchholz: Commun. Math. Phys. 36, 287-304 (1974; Zbl 0289.46050)] by using the zeroth component of a conserved current for the field \(\phi(f)\otimes 1+1\otimes \phi(f)\) and explicitly constructing the unitary in \(\tilde {\mathfrak B}\) implementing the flip automorphism of \(\tilde {\mathfrak A}\).
Reviewer: H.Araki

MSC:
46L35 Classifications of \(C^*\)-algebras
46L60 Applications of selfadjoint operator algebras to physics
81T05 Axiomatic quantum field theory; operator algebras
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