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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.

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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