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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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