We answer a question of E. Kirchberg (personal communication): does the relative commutant of a separable C*-algebra in its ultrapower depend on the choice of the ultrafilter? All algebras and all subalgebras in this note are C*-algebras and C*subalgebras, respectively, and all ultrafilters are nonprincipal ultrafilters on N. Our C*-terminology is standard (see e.g., [2]). In the following U ra...

A C* algebra is a closed self-adjoint subalgebra of B(H), the bounded operators on a Hilbert space H. Kirchberg asked whether every C* algebra embeds into an ultrapower of a particular C* algebra, the Cuntz algebra O_2. In this talk, I will explain the model-theoretic content of this problem. In particular, I will show how model-theoretic forcing can be used to give a weak local reformulation o...

The Arens products are the standard way of extending the product from a Banach algebra A to its bidual A′′. Ultrapowers provide another method which is more symmetric, but one that in general will only give a bilinear map, which may not be associative. We show that if A is Arens regular, then there is at least one way to use an ultrapower to recover the Arens product, a result previously known ...

We study several properties and applications of the ultrapower MU a metric space M. prove that Lipschitz-free F(MU) is finitely representable in F(M). also characterize spaces are Lipschitz Banach as those biLipschitz embed into an space. Thanks to this link, we obtain if M X, then F(M) F(X). apply these results cotype stability functions under ultraproducts.

Eliasson, J. 2003: Ultrasheaves. Uppsala dissertations in Mathematics 30. 54 pp. Uppsala. ISBN 91-506-1716-8. This thesis treats ultrasheaves, sheaves on the category of ultrafilters. In the classical theory of ultrapowers, you start with an ultrafilter (I,U) and, given a structure S, you construct the ultrapower S/U . The fundamental result is ! Loś’s theorem for ultrapowers giving the connect...

We present a uniform version of Di Nola Theorem, this enables to embed all MV-algebras of a bounded cardinality in an algebra of functions with values in a single non-standard ultrapower of the real interval [0, 1]. This result also implies the existence, for any cardinal α, of a single MValgebra in which all infinite MV-algebras of cardinality at most α embed. Recasting the above construction ...

We present two constructions of models of bounded arithmetic, both in the form of a generalization of the ultrapower construction, that yield nonelementary extensions but do not introduce new lengths. As an application we show, assuming the existence of a one-way permutation g hard against polynomial-size circuits, that strictR 2(g) is weaker than R 2(g). In particular, if such a permutation ca...

This paper explores several topics related to Woodin’s HOD conjecture. We improve the large cardinal hypothesis of dichotomy theorem from an extendible a strongly compact cardinal. show that assuming there is and holds, no elementary embedding HOD, settling question Woodin. equivalent uniqueness property embeddings levels cumulative hierarchy. prove holds if only every regular above first carri...