Logic and operator algebras

The most recent wave of applications of logic to operator algebras is a young and rapidly developing field. This is a snapshot of the current state of the art.Comment: A minor chang

Model theory of operator algebras III: Elementary equivalence and II_1 factors

We use continuous model theory to obtain several results concerning isomorphisms and embeddings between II_1 factors and their ultrapowers. Among other things, we show that for any II_1 factor M, there are continuum many nonisomorphic separable II_1 factors that have an ultrapower isomorphic to an ultrapower of M. We also give a poor man's resolution of the Connes Embedding Problem: there exists a separable II_1 factor such that all II_1 factors embed into one of its ultrapowers.Comment: 16 page

A survey on the model theory of tracial von Neumann algebras

We survey the developments in the model theory of tracial von Neumann algebras that have taken place in the last fifteen years. We discuss the appropriate first-order language for axiomatizing this class as well as the subclass of II$_1$ factors. We discuss how model-theoretic ideas were used to settle a variety of questions around isomorphism of ultrapowers of tracial von Neumann algebras with respect to different ultrafilters before moving on to more model-theoretic concerns, such as theories of II$_1$ factors and existentially closed II$_1$ factors. We conclude with two recent applications of model-theoretic ideas to questions around relative commutants.Comment: 27 pages; first draft; comments welcome; to appear in the volume "Model theory of operator algebras" as part of DeGruyter's Logic and its Application Serie

Some interesting problems

A ≤W B. (This refers to Wadge reducible.) Answer: The first question was answered by Hjorth [83] who showed that it is independent. 1.2 A subset A ⊂ ω ω is compactly-Γ iff for every compact K ⊂ ω ω we have that A ∩ K is in Γ. Is it consistent relative to ZFC that compactly-Σ 1 1 implies Σ 1 1? (see Miller-Kunen [111], Becker [11]) 1.3 (Miller [111]) Does ∆ 1 1 = compactly- ∆ 1 1 imply Σ 1 1 = compactly-Σ 1 1? 1.4 (Prikry see [62]) Can L ∩ ω ω be a nontrivial Σ 1 1 set? Can there be a nontrivial perfect set of constructible reals? Answer: No, for first question Velickovic-Woodin [192]. question Groszek-Slaman [71]. See also Gitik [67]

Algebraic nonlinear collective motion

Finite-dimensional Lie algebras of vector fields determine geometrical collective models in quantum and classical physics. Every set of vector fields on Euclidean space that generates the Lie algebra sl(3, R) and contains the angular momentum algebra so(3) is determined. The subset of divergence-free sl(3, R) vector fields is proven to be indexed by a real number $\Lambda$. The $\Lambda=0$ solution is the linear representation that corresponds to the Riemann ellipsoidal model. The nonlinear group action on Euclidean space transforms a certain family of deformed droplets among themselves. For positive $\Lambda$, the droplets have a neck that becomes more pronounced as $\Lambda$ increases; for negative $\Lambda$, the droplets contain a spherical bubble of radius $|\Lambda|^{{1/3}}$. The nonlinear vector field algebra is extended to the nonlinear general collective motion algebra gcm(3) which includes the inertia tensor. The quantum algebraic models of nonlinear nuclear collective motion are given by irreducible unitary representations of the nonlinear gcm(3) Lie algebra. These representations model fissioning isotopes ($\Lambda>0$) and bubble and two-fluid nuclei ($\Lambda<0$).Comment: 32pages, 4 figures not include