Isaac Goldbring

We study expansions of Hilbert spaces with a bounded normal operator T . We axiomatize this theory in a natural language and identify all of its completions. We prove the definability of the adjoint T ∗ $T^*$ upper T Superscript asterisk and prove quantifier elimination for every completion after adding T ∗ $T^*$ upper T Superscript asterisk to the language. We identify types with measures on the spectrum of the operator and show that the logic topology on the type space corresponds to the weak*-topology on the space of measures. We also give a precise formula for the metric on the space of 1 $1$ 1 -types. We prove all completions are stable and characterize the stability spectrum of the theory in terms of the spectrum of the operator. We also show all completions, regardless of their spectrum, are ω $\omega $ omega -stable up to perturbations.

We give an example of a unital C ∗ $\mathrm {C}^*$ normal upper C Superscript asterisk -algebra A $\mathbf {A}$ bold upper A with a computable presentation and for which neither K 0 ( A ) $K_0(\mathbf {A})$ upper K 0 left parenthesis bold upper A right parenthesis nor K 1 ( A ) $K_1(\mathbf {A})$ upper K 1 left parenthesis bold upper A right parenthesis has a computable presentation.

We continue the study of the effective content of $K$-theory for C*-algebras, with a focus on AF algebras. We show that from a c.e. presentation of an AF algebra it is possible to compute a representation of the algebra as an inductive limit of finite-dimensional algebras. Using this, and an analogous result for dimension groups, we show that the computable $K_0$ functor provides a computable equivalence of categories between c.e. presentations of AF algebras and c.e. presentations of unital (scaled) dimension groups, giving an effective version of Elliott's classification theorem. We use our results to determine the complexity of the index set and isomorphism problems for various classes of AF algebras.

We initiate the study of definability (in the model-theoretic sense) of C*-tensor norms. We show that neither the minimal nor maximal tensor norms are definable uniformly over all C*-algebras. The proof in the case of the minimal tensor product norm uses a deep theorem of Kirchberg characterizing exactness in terms of tensor products with matrix ultraproducts while the case of maximal tensor products uses Pisier's recent characterization of the lifting property in terms of maximal tensor products and ultraproducts. We also study the question of when one of these tensor products can be definable in a particular C*-algebra. We establish some negative results along these lines for particular C*-algebras and when the definability condition is strengthened to be computable and of a restricted quantifier-complexity; these results use the quantum complexity results MIP$^*$=RE and MIP$^{co}=$coRE. As a byproduct of our arguments, we answer a question of Fritz, Netzer, and Thom by showing that the norm on $C^*(\mathbb{F}_n\times \mathbb{F}_n)$ is not computable for any $n\in \{2,3,\ldots,\infty\}$.

We study expansions of Hilbert spaces with a bounded normal operator $T$. We axiomatize this theory in a natural language and identify all of its completions. We prove the definability of the adjoint $T^*$ and prove quantifier elimination for every completion after adding $T^*$ to the language. We identify types with measures on the spectrum of the operator and show that the logic topology on the type space corresponds to the weak*-topology on the space of measures. We also give a precise formula for the metric on the space of $1$-types. We prove all completions are stable and characterize the stability spectrum of the theory in terms of the spectrum of the operator. We also show all completions, regardless of their spectrum, are $ω$-stable up to perturbations.

Given a countable group $Γ$, letting $\mathcal{K}_Γ$ denote the class of {\pmp} actions of $Γ$, we study the question of when the model companion of $\mathcal{K}_Γ$ exists. Berenstein, Henson, and Ibarlucía showed that the model companion of $\mathcal{K}_Γ$ exists when $Γ$ is a nonabelian free group on a countable number of generators. We significantly generalize their result by showing that the model companion of $\cal K_Γ$ exists whenever $Γ$ is an approximately treeable group. The class of approximately treeable groups contain the class of treeable groups as well as the class of universally free groups, that is, the class of groups with the same universal theory as nonabelian free groups. We prove this result using an open mapping characterization of when the model companion exists; moreover, this open mapping characterization provides concrete, ergodic-theoretic axioms for the model companion when it exists. We show how to simplify these axioms in the case of treeable groups, providing an alternate axiomatization for the model companion in the case of the free group, which was first axiomatized by Berenstein, Henson, and Ibarlucía using techniques from model-theoretic stability theory. Along the way, we prove a purely ergodic-theoretic result of independent interest, namely that finitely generated universally free groups (also known as limit groups) have Kechris' property MD. We also show that for groups with Kechris' EMD property, the profinite completion action is existentially closed, and for groups without property (T), the generic existentially closed action is weakly mixing, generalizing results of Berenstein, Henson, and Ibarlucía for the case of nonabelian free groups.