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\(\mathrm{II}_1\) factors with nonisomorphic ultrapowers. (English) Zbl 1384.46040
Summary: We prove that there exist uncountably many separable \(\mathrm{II}_{1}\) factors whose ultrapowers (with respect to arbitrary ultrafilters) are nonisomorphic. In fact, we prove that the families of nonisomorphic \(\mathrm{II}_{1}\) factors originally introduced by McDuff are such examples. This entails the existence of a continuum of nonelementarily equivalent \(\mathrm{II}_{1}\) factors, thus settling a well-known open problem in the continuous model theory of operator algebras.

MSC:
46L36 Classification of factors
46L10 General theory of von Neumann algebras
03C20 Ultraproducts and related constructions
46M07 Ultraproducts in functional analysis
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