arXiv subjects
Publications and source records attributed to Bradd Hart.
We realize the Jiang-Su algebra, all UHF algebras, and the hyperfinite II$_{1}$ factor as Fraïssé limits of suitable classes of structures. Moreover by means of Fraïssé theory we provide new examples of AF algebras with strong homogeneity properties. As a consequence of our analysis we deduce Ramsey-theoretic results about the class of full-matrix algebras.
We begin the study of categorical logic for continuous model theory. In particular, we 1. introduce the notions of metric logical categories and functors as categorical equivalents of a metric theory and interpretations, 2. prove a continuous version of conceptual completeness showing that $T^\eq$ is the maximal conservative expansion of $T$, and 3. define the concept of a metric pre-topos.
The relative commutant $A'\cap A^{\mathcal{U}}$ of a strongly self-absorbing algebra $A$ is indistinguishable from its ultrapower $A^{\mathcal{U}}$. This applies both to the case when $A$ is the hyperfinite II$_1$ factor and to the case when it is a strongly self-absorbing C*-algebra. In the latter case we prove analogous results for $\ell_\infty(A)/c_0(A)$ and reduced powers corresponding to other filters on $\bf N$. Examples of algebras with approximately inner flip and approximately inner half-flip are provided, showing the optimality of our results. We also prove that strongly self-absorbing algebras are smoothly classifiable, unlike the algebras with approximately inner half-flip.
Recently, Boutonnet, Chifan, and Ioana proved that McDuff's family of continuum many pairwise nonisomorphic separable II$_1$ factors are in fact pairwise non-elementarily equivalent by proving that any ultrapowers of two distinct members of the family are nonsiomorphic. We use Ehrenfeucht-Fraisse games to provide an upper bound on the quantifier-depth of sentences which distinguish these theories.
We show that the excellence axiom in the definition of Zilber's quasiminimal excellent classes is redundant, in that it follows from the other axioms. This substantially simplifies a number of categoricity proofs.
We examine the properties of existentially closed (R^omega-embeddable) II_1 factors. In particular, we use the fact that every automorphism of an existentially closed (R^omega-embeddable) II_1 factor is approximately inner to prove that Th(R) is not model-complete. We also show that Th(R) is complete for both finite and infinite forcing and use the latter result to prove that there exist continuum many nonisomorphic existentially closed models of Th(R).
The Connes Embedding Problem (CEP) asks whether every separable II_1 factor embeds into an ultrapower of the hyperfinite II_1 factor. We show that the CEP is equivalent to the computability of the universal theory of every type II_1 von Neumann algebra. We also derive some further computability-theoretic consequences of the CEP.
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.
We observe how a classical model-theoretic fact proves the existence of many strong amalgamation bases for the class of R^ω-embeddable von Neumann algebras, where R is the hyperfinite II_1 factor. In particular, we shows that R itself is a strong amalgamation base.
We give model-theoretic characterizations of UHF algebras and of AF algebras as C*-algebras that omit certain sets of types.
We present unified proofs of several properties of the corona of $σ$-unital C*-algebras such as AA-CRISP, SAW*, being sub-$σ$-Stonean in the sense of Kirchberg, and the conclusion of Kasparov's Technical Theorem. Although our results were obtained by considering C*-algebras as models of the logic for metric structures, the reader is not required to have any knowledge of model theory of metric structures (or model theory, or logic in general). The proofs involve analysis of the extent of model-theoretic saturation of corona algebras.
Several authors have considered whether the ultrapower and the relative commutant of a C*-algebra or II_1 factor depend on the choice of the ultrafilter. We settle each of these questions, extending results of Ge-Hadwin and the first author.
In this note, we show that the theory of tracial von Neumann algebras does not have a model companion. This will follow from the fact that the theory of any locally universal, McDuff II_1 factor does not have quantifier elimination. We also show how a positive solution to the Connes Embedding Problem implies that there can be no model-complete theory of 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.
Let T be a complete, first-order theory in a finite or countable language having infinite models. Let I(T,kappa) be the number of isomorphism types of models of T of cardinality κ. We denote by μ(respectively \hatμ) the number of cardinals (respectively infinite cardinals) less than or equal to κ. We prove that I(T,κ), as a function of κ> \aleph_0, is the minimum of 2^κ and one of the following functions: 1. 2^κ; 2. the constant function 1; 3. |\hatμ^n/{\sim_G}|-|(\hatμ- 1)^n/{\sim_G}| if \hatμ<ωfor some 1 = ωsome group G <= Sym(n); 4. the constant function \beth_2; 5. \beth_{d+1}(μ) for some infinite, countable ordinal d; 6. \sum_{i=1}^d Γ(i) where d is an integer greater than 0 (the depth of T) and Γ(i) is either \beth_{d-i-1}(μ^{\hatμ}) or \beth_{d-i}(μ^{σ(i)} + α(i)), where σ(i) is either 1, \aleph_0 or \beth_1, and α(i) is 0 or \beth_2; the first possibility for Γ(i) can occur only when d-i > 0.
We present a general framework for carrying out some constructions. The unifying factor is a combinatorial principle which we present in terms of a game in which the first player challenges the second player to carry out constructions which would be much easier in a generic extension of the universe, and the second player cheats with the aid of Diamond. Section 1 contains an axiomatic framework suitable for the description of a number of related constructions, and the statement of the main theorem in terms of this framework. In section 2 we illustrate the use of our combinatorial principle. The proof of the main result is then carried out in sections 3-5.
Suppose L is a relational language and P in L is a unary predicate. If M is an L-structure then P(M) is the L-structure formed as the substructure of M with domain {a: M models P(a)}. Now suppose T is a complete first order theory in L with infinite models. Following Hodges, we say that T is relatively lambda-categorical if whenever M, N models T, P(M)=P(N), |P(M)|= lambda then there is an isomorphism i:M-> N which is the identity on P(M). T is relatively categorical if it is relatively lambda-categorical for every lambda. The question arises whether the relative lambda-categoricity of T for some lambda >|T| implies that T is relatively categorical. In this paper, we provide an example, for every k>0, of a theory T_k and an L_{omega_1 omega} sentence varphi_k so that T_k is relatively aleph_n-categorical for n < k and varphi_k is aleph_n-categorical for n<k but T_k is not relatively beth_k-categorical and varphi_k is not beth_k-categorical.