Operator Algebras and Operator Theory Ical Atom

In a joint work with Tattwamasi Amrutam and Eli Glasner, we study intermediate $C^*$-algebras of the form $C^*_r(\Gamma) < \mathcal{A} < C(X) \rtimes \Gamma$, where $\Gamma \curvearrowright X$ is a given minimal action of a countable discrete group $\Gamma$ on a compact space $X$. Every $\Gamma$-factor of the given topological dynamical system $X \rightarrow Y$ gives rise to an intermediate algebra of the form $\mathcal{A} = C(Y) \rtimes \Gamma$, and by analogy we may think of more general factors as representing ‘‘non-abelian’’ factors. Let us call the dynamical system ``reflecting’’ if the only intermediate algebras come from dynamical factors.

We show that another source of intermediate algebras comes from ideals in $C^*_r(\Gamma)$. In particular, we show that if $\Gamma$ is not $C^*$-simple, $X$ admits a $\Gamma$-invariant probability measure, and the cardinality of $X$ is at least $3$, then the system is not reflecting.

In the talk, I will focus on the example highlighted in the title. In this case, we obtain a complete description of all intermediate algebras in terms of some combinatorial data described in terms of ideals in $C^*_r(\mathbb{Z})$. In particular there are uncountably many intermediate algebras, as compared to only countably many dynamical factors. I will show how our description can often be used in order to obtain structural information about the algebras, such as simplicity, the existence of a center, and a closed formula for the algebra generated by two given ones.

The noncommutative (nc) disc algebra $\mathcal{A}_d$ was studied extensively first by Popescu. It is the norm closed operator algebra generated by the left creation operators on the full Fock space. This algebra is semi-Dirichlet. Namely, $\mathcal{A}_d^* \mathcal{A}_d \subset \overline{\mathcal{A}_d + \mathcal{A}_d^*} = \mathcal{S}_d$. Therefore, one can perform a GNS type construction to obtain representations of $\mathcal{A}_d$ from states on $\mathcal{S}_d.$ This observation is one of the ingredients in the nc Clark theory developed by Jury and Martin.

In this talk, I will focus on nc rational functions and, in particular, inner ones. I will show how one obtains from an nc inner rational a finitely-correlated state (Bratteli and Jorgensen) on the Cuntz algebra. Connect the finitely-correlated states to minimal isometric dilations of finite-dimensional row coisometries and the work of Davidson, Kribbs, and Shpigel. Lastly, I will show that many finitely-correlated states are peak states in the sense of Clouatre and Thomson.

This talk is based on joint works with Mike Jury and Rob Martin.

In a joint work with Yair Hartman and Hanna Oppelmayer, we study the sub-von Neumann Algebras of the group von Neumann algebra $L\Gamma$. We will first show that $L\Gamma$ admits a maximal invariant amenable subalgebra. We will also introduce the notion of invariant probability measures on the space of sub-von Neumann algebras (IRAs) which is analogous to the concept of Invariant Random Subgroups. We shall show that amenable IRAs are supported on the maximal amenable invariant subalgebra.

The talk will be aimed at an audience which is not necessarily familiar with the concepts below, and I’ll aim to explain them at the expense of providing proofs. The Elliott classification program for simple nuclear C-algebras reached an important milestone in the past decade. This program aims to classify simple separable nuclear C-algebras in terms of the Elliott invariant, consisting of the K-theory groups and tracial data. It is now established that such C-algebras can be classified provided they are Z-stable (a key regularlity condition) and satisfy the Rosenberg-Schochet Universal Coefficient Theorem (where it is a major open problem to determine whether this theorem holds for all simple nuclear separable C-algebras). The classification theorem was shown not to extend beyond the Z-stable case. Specifically, Toms constructed examples of simple separable nuclear unital C*-algebras with the same Elliott invariant, but which can be distinguished by another invariant he called the radius of comparison, which measures to what extent positive elements fail to be compared via tracial data. While there were good reasons to think that simply adding the radius of comparison to the invariant would not be sufficient to extend the classification theorem further, intriguingly, recently, Elliott, Li and Niu proved a classification theorem for so-called Villadsen-type algebras using the combination of the Elliott invariant and the radius of comparison, an invariant which was introduced by Toms in order to distinguish between certain non-isomorphic AH algebras with the same Elliott invariant. This might have raised the prospect that the Elliott classification program can be extended beyond the Z-stable case by adding the radius of comparison to the invariant. I will discuss a recent preprint in which we show that this is not the case: we construct an uncountable family of non-isomorphic AH algebras with the same Elliott and same radius of comparison. We can distinguish between them using a finer invariant, which we call the local radius of comparison. This is joint work with N. Christopher Phillips.

In this talk, I will discuss isometric dilations of completely contractive representations (in short c.c. representation) of product systems (of $W^∗$-correspondences) over the semigroup $\mathbb{Z}^n_{+}$. It is known that for $n = 1, 2$, c.c. representations of such product systems always have isometric dilations and the result fails for $n > 2$, in general. We will see that under certain positivity and pureness conditions c.c. representations of product systems over $\mathbb{Z}^n_{+}$ have isometric dilations, also we will see an explicit form of the dilations. If time permits, I will discuss some applications of it.

This talk is based on joint work with Monojit Bhattacharjee and Baruch Solel.

Self-testing is the strongest form of quantum functionality verification, which allows one to deduce the quantum state and measurements of an entangled system from its classically observed statistics. From a mathematical perspective (which will be the perspective of this talk), self-testing is an intriguing uniqueness phenomenon, pertaining to functional analysis, moment problems, convexity and representation theory. This talk addresses basic motivation and ideas behind self-testing, and discusses which states and measurements can be self-tested. In particular, the talk focuses on how tuples of projections adding to a scalar multiple of identity, and Jordan algebras find its way into this corner of quantum information theory. Based on joint work with Ranyiliu Chen and Laura Mančinska.

Given a $d$-dimensional ($d < \infty$) operator space $\mathcal{E}$ with basis $\{Q_1, \cdots, Q_d\}$, consider the corresponding noncommutative (nc) operator ball $\mathbb{D}_Q := \{ X \in \mathbb{M}^d : \| \sum_j Q_j \otimes X_j \| < 1 \}$. In this talk, we discuss the problem of extending certain biholomorphic maps between subvarieties $\mathfrak{V}_1$ and $\mathfrak{V}_2$ of nc operator balls $\mathbb{D}_{Q^{(1)}}$ and $\mathbb{D}_{Q^{(2)}}$.

For trivial reasons, such an extension cannot exist in general, and we discuss several examples to showcase the obstructions. When the operator spaces $\mathcal{E}^{(1)}$ and $\mathcal{E}^{(2)}$ are both injective, and the subvarieties $\mathfrak{V}_1$ and $\mathfrak{V}_2$ are both homogeneous, we show that a biholomorphism between $\mathfrak{V}_1$ and $\mathfrak{V}_2$ can be extended to a biholomorphism between $\mathbb{D}_{Q^{(1)}}$ and $\mathbb{D}_{Q^{(2)}}$. Moreover, we show that if such an extension exists then there exists a linear isomorphism between $\mathbb{D}_{Q^{(1)}}$ and $\mathbb{D}_{Q^{(2)}}$ that sends $\mathfrak{V}_1$ to $\mathfrak{V}_2$.

Finite Rokhlin dimension, a generalization of the Rokhlin property, is a regularity property for actions of certain groups on C-algebras. The main interest in Rokhlin dimension was its use to establish various permanence properties: for example, if the C-algebra acted on has finite nuclear dimension and the action has finite Rokhlin dimension then the crossed product again has finite nuclear dimension. As such, the main interest in Rokhlin dimension was to show that it is finite, and not much attention was paid to its actual value. In particular, while it is known that there are actions with positive finite Rokhlin dimension (that is, have finite Rokhlin dimension but do not have the Rokhlin property, which corresponds to Rokhlin dimension zero), there were no examples of actions of finite groups with finite Rokhlin dimension greater than 2. I’ll discuss a recent preprint in which we provide examples of actions of finite groups on simple AF algebras with arbitrarily large finite Rokhlin dimension. This shows that Rokhlin dimension is not just a tool to establish regularity results, but is an interesting invariant for group actions, which in a sense measures the complexity of the action.

This is joint work with N. Christopher Phillips.