No abstract

We introduce a version of logic for metric structures suitable for applications to C*-algebras and tracial von Neumann algebras. We also prove a purely model-theoretic result to the effect that the theory of a separable metric structure is stable if and only if all of its ultrapowers associated with nonprincipal ultrafilters on N are isomorphic even when the Continuum Hypothesis fails. norm) unit ball of M. The completion of M with respect to this metric is isomorphic to a Hilbert space (see, e.g., [4] or [17]).The algebra of all sequences in M bounded in the operator norm is denoted by ℓ ∞ (M). If U is an ultrafilter on N thenis a norm-closed two-sided ideal in ℓ ∞ (M), and the tracial ultrapower M U (also denoted by U M) is defined to be the quotient ℓ ∞ (M)/c U . It is well-known that M U is tracial, and a factor if and only if M is-see, e.g., [4] or [24]; this also follows from axiomatizability ( §3.2) and Loś's theorem (Proposition 4.3 and the remark afterwards).Elements of M U will either be denoted by boldface Roman letters such as a or represented by sequences in ℓ ∞ (M). Identifying a tracial von Neumann algebra M with its diagonal image in M U , we will also work with the relative commutant of M in its ultrapower,

In this note we give an answer to the following problem of Todorcevic: Find out the combinatorial essence behind the fact that the family JP of the ground-model infinite sets of integers in a Perfect-set forcing extension has the property that for any Borel/: [N] a ->{0, 1} there exists an AeJt such that/ is constant on [A] m (see [7], [13]). In other words, one needs to capture the combinatorial properties of the family Jf of ground-model subsets of M which assure that it diagonalizes all Borel partitions. It turns out that the notion which results from our analysis of this problem is a bit more optimal than the older notion of a "happy family" (or selective coideal) introduced by A.R.D. Mathias [16] long ago in order to extend the well-known theorems of Galvin-Prikry [6] and Silver [25] (see Theorems 3.1 and 4.1 below). We should remark that these Mathias-style extensions can indeed be as useful in the applications as the original partition theorems. For example, one such application (where the original partition theorem of Galvin Prikry and Silver does not seem to fit) was recently found by Todorcevic ([28]) in order to supply a new proof of the famous Bourgain Fremlin-Talagrand theorem ([2]). Other applications can be found in the so-called parametrized partition calculus (see e.g., [17], [19], [29], [38]). One can also use these Mathias-style extensions of the Galvin Prikry and Silver theorems to give a new proof of the well-known perfect-tree theorem of J. Stern ([39], see also [38, §C]).

Several authors have considered whether the ultrapower and the relative commutant of a C*‐algebra or II1 factor depend on the choice of the ultrafilter. We settle each of these questions, extending results of Ge–Hadwin and the first author.

We prove that it is relatively consistent with the usual axioms of mathematics that all automorphisms of the Calkin algebra are inner. Together with a 2006 Phillips-Weaver construction of an outer automorphism using the Continuum Hypothesis, this gives a complete solution to a 1977 problem of Brown-Douglas-Fillmore. We also give a simpler and self-contained proof of the Phillips-Weaver result.