Operator Algebras Ical Atom

Finite Rokhlin dimension is one of the several ways in which the Rokhlin property, a concept originally generalized from ergodic theory to the framework of amenable actions on von Neumann algebras, has been adapted to $C^\ast$-dynamics. A nice feature of the notion of finite Rokhlin dimension is that, although it has weaker requirements compared to other adaptations of the Rokhlin property to actions on $C^\ast$-algebras, it still induces useful regularity properties on the actions satisfying it. For instance, finite nuclear dimension (a non-commutative generalization of the notion of topological covering dimension) and Z-stability (Z is the Jiang-Su algebra) are preserved when taking the crossed product of a separable unital $C^\ast$-algebra by an action of the integers which has finite Rokhlin dimension. In this talk I’ll show that for a finite group action alpha on a separable, simple, unital, Z-stable, nuclear $C^\ast$-algebra A with non-empty trace space, the action alpha is strongly outer if and only if alpha tensor the identity on Z has finite Rokhlin dimension. The novelty of this result is that we make no topological assumption on the trace space of A, in opposition to past works proving analogous statements, where the trace space is always assumed to be a Bauer simplex. This is a joint work with Ilan Hirshberg.

Joint work with Fernando Abadie and Eusebio Gardella.

We introduce and study a property, called “the decomposition property”, for $C^*$-algebraic partial actions by discrete groups (we will focus on finite groups in this talk). Partial actions with this property (“decomposable partial actions”) behave in many ways like global actions, which makes their study particularly accessible. We show that a decomposable partial action always admits a globalization, which we describe explicitly. Moreover, we carry out a complete description of crossed products by decomposable partial actions as a direct sum of crossed products by globally acting finite groups. Finally, we show that decomposable partial actions appear naturally in practice. For example, every partial action by a finite group is an iterated extension of decomposable partial actions. Using this, we derive (for finite groups) a full description of the partial group $C^*$-algebra as a direct sum of matrix algebras over (global) group $C^*$-algebras. This description (using different machinery) appeared in a paper by Dockuchaev, Exel, and Piccione.

I will report on joint work in progress with N. Christopher Phillips.

In 2010, Giol and Kerr published a construction of a minimal dynamical system whose associated crossed product has positive radius of comparison. Subsequently, Phillips and Toms conjectured that the radius of comparison of a crossed product should be roughly half the mean dimension of the underlying system. Upper bounds were obtained by Phillips, Hines-Phillips-Toms and very recently by Niu, however there were no results concerning lower bounds aside for the examples of Giol and Kerr. In the non-dynamical context, work of Elliott and Niu suggests that the right notion of dimension to consider is cohomological dimension, rather than covering dimension (notions which coincide for CW complexes). Motivated by this insight, we introduce an invariant which we call “mean cohomological independence dimension” (more precisely, a sequence of such invariants), for actions of countable amenable groups on compact metric spaces, which are related to mean dimension, and obtain lower bounds for the radius of comparison for crossed products in terms of this invariant.

In this talk, I will present joint work with Matt Kennedy. Choquet simplifies are infinite-dimensional versions of classical simplices. They arise naturally as the collections of all probability Borel measures on a compact Hausdorff space and as the collections of invariant measures of a dynamical system. Namioka and Phelps characterized Choquet simplices via nuclearity of the associated functions system. In the non commutative generalisation one replaces function systems with operator systems and non commutative convex sets with matrix/nc convex sets. In this talk, I will define nc Choquet simplices, as well as nc analogs of Bauer and Poulsen simplices. It turns out, that as in the classical case, every nc Bauer simplex is the nc state space of a C^*-algebra. Lastly, I will discuss dynamical applications, in particular a non commutative version of a theorem of Glasner and Weiss.

There is a general approach that emerged in several different areas of mathematics over last several decades for passing from the commutative setting to the free noncommutative setting. This approach that is sometimes referred to as quantization replaces the original object of study (e.g., a vector space) by square matrices of all sizes over this object. A notable example in the area of functional analysis is the theory of operator algebras, operator systems, and operator spaces. Another example is the so called free noncommutative function theory that originated in the pioneering work of J.L. Taylor on noncommutative spectral theory in the 1970s (as well as the earlier work of Takesaki on the noncommutative version of Gelfand’s theory for $C^*$ algebras) and was vigorously developed in recent years.

In this talk I will discuss completely positive free noncommutative (cp nc) kernels which are the analogue in the setting of free noncommutative function theory of the usual positive kernel functions that played a prominent role in complex analysis and operator theory since the foundational work of Aronszajn, and that also generalize completely positive maps as well as more recent completely positive kernels of Barreto–Bhat–Liebscher–Skeide. More specifically I will discuss three problems: (a) extending the values of a cp nc kernel from maps defined on an operator system to maps defined on a $C^*$ algebra (the Arveson extension); (b) representing a completely bounded hermitian free noncommutative kernel as a linear combination of cp nc kernels (the Jordan decomposition); (c) giving certificates for one hermitian kernel to be positive at all the points where another one is positive (analogously to the Positivstellensaetze of commutative and noncommutative real algebraic geometry).

This is a joint work with Joe Ball and Gregory Marx.

See attached

In my talk I will present several results whose unifying theme is a matrix-valued analogue of the numerical range, called the matrix range of an operator tuple. After explaining what is the matrix range and what it is good for, I will report on recent work in which we prove that there is a certain “universal” matrix range, to which the matrix ranges of a sequence of large random matrices tends to, almost surely. The key novel technical aspects of this work are the (levelwise) continuity of the matrix range of a continuous field of operators, and a certain quantitative matrix valued Hahn-Banach type separation theorem. In the last part of the talk I will explain how the (uniform) distance between matrix ranges can be interpreted equivalently as a “dilation distance”, which can be interpreted as a kind of “representation distance”. These vague ideas will be illustrated with an application: the construction of a norm continuous family of representations of the noncommutative tori (recovering a result of Haagerup-Rordam in the d=2 case and of Li Gao in the d>2 case).

Based on joint works with Malte Gerhold, Satish Pandey and Baruch Solel.