We obtain a new, coordinate free, characterization of quasidiagonal operators with essential spectra contained in the unit circle by adapting the proof of a classical result in the theory of Banach spaces.

In the operator algebraic theory of compact quantum groups, a given compact quantum group 𝔾 can often be described by more than one C*-algebra of "functions" on 𝔾. For example, one has a universal (or full) C*-algebra as well as a reduced C*-algebra associated to 𝔾. Any C*-algebraic description that appears as a non-trivial intermediate quotient between the full and reduced C*-algebras is considered exotic. Surprisingly, finding interesting examples of exotic compact quantum group C*-algebras is rather difficult. In this talk, I will introduce the notion of a unitary representation of a unimodular discrete quantum group Γ associated the non-commutative L_p-space of Γ. We will then show how the theory of L_p-representations together with Pontryagin duality can be used to construct new examples of exotic compact quantum group C*-algebras for certain examples of free quantum groups. This is joint work with Zhong-Jin Ruan.

If Γ is a non-type I discrete group, there is essentially no hope of a reasonable characterization of its irreducible representations up to unitary equivalence. If one instead focuses on only the C*-algebras generated by irreducible representations of Γ the situation is a little more promising. Recent events in the theory of C*- algebras suggest that–in the case Γ is a finitely generated torsion free nilpotent group–ordered K-theory may serve as a complete invariant for the C*-algebras generated by irreducible representations of Γ. In this talk we will focus on quasidiagonality of unitary representations of nilpotent groups, the regularity properties of the C*-algebras generated by these representations and how this feeds into the larger goal of characterizing C*-algebras generated by irreducible representation of nilpotent groups.

For a discrete group G, we consider the minimal C*-algebra of ℓ^∞(G) that arises as the image of a unital positive G-equivariant projection. This algebra always exists and is unique up to isomorphism. It is trivial if and only if G is amenable. It turns out that, more generally, it can be identified with the C*-algebra of continuous functions on Furstenberg’s universal G-boundary. This construction leads to a proof of a conjecture of Ozawa about nuclear embeddings of reduced C*-algebras of exact groups, and has a number of other interesting consequences. This is joint work with Mehrdad Kalantar (Carleton University).

The chromatic number of a graph has a description as the value of a three-person game. If this game is allowed to be played with quantum probabilities, then this yields a smaller value--the quantum chromatic number. However, there are at least three possible convex sets that could represent these quantum probabilities depending on whether or not conjectures of Connes and Tsirelson have positive answers. Thus, we introduce and study three possible quantum chromatic and study their properties. Proving that these integers are all equal would give a partial affirmation of these conjectures, any inequalities would yield a counterexample to one or more conjectures.

This is an account of joint work with N. Ozawa. For any pair M,N of von Neumann algebras such that the algebraic tensor product M⊗N admits more than one C*-norm, the cardinal of the set of C*-norms is at least 2^ℵ₀. Moreover there is a family with cardinality 2^ℵ₀ of injective tensor product functors for C*-algebras in Kirchberg's sense. Let B=∏_n M_n. We also show that, for any non-nuclear von Neumann algebra M⊂B(ℓ₂), the set of C*-norms on B⊗M has cardinality equal to 2^2ℵ₀. The talk will also recall the connection of such questions with the non-separability of the set of finite dimensional (actually 3-dimensional) operator spaces which goes back to a 1995 paper with Marius Junge, and several recent "quantitative" refinements obtained using quantum expanders.

A separable real Banach space X has property β [Rolewicz, 1987] if for every ε>0 one can find δ>0 such that for every x∈X, ∥x∥<1+δ implies α[conv({x}∪B_X)\B_X]<ε, where α is the Kuratowski measure of noncompactness, and B_X is the unit ball of X. Rolewicz introduced property β as a generalization of uniform convexity. In fact, a Banach space X is uniformly convex if and only if the above property is satisfied when the measure of noncompactness α is replaced by diameter. In this talk, we will trace the isometric and isomorphic characterizations of property β that are present in the literature. We will then present the application of property β is the study of nonlinear quotient maps. In particular, we will focus on a joint work with Dilworth and Kutzarova where we show, via the construction of a Laakso-type graph, that the isomorphic class of separable Banach spaces with property β is closed under taking uniform quotient maps. If time permits, we will talk about the quantitative version of this question as well.

The talk will deal with algebras which arise from commutants modulo certain normed ideals of n-tuples of operators and their quotients by their ideal of compact operators. This will include duality properties, K-theory aspects and the model theory property of degree-1 saturation.

Discovered by Einstein, Podolsky and Rosen in 1935, quantum entanglement has become the key ingredient for modern quantum information theory. However, over 60 years ago, even Einstein himself could not believe this strange quantum entanglement phenomenon and called it “spooky action at a distance”, because it is so difficult to detect quantum entanglement. In this talk, we will discuss the typicality of quantum entanglement. Our model for random induced quantum states are random matrices. We show that there is a threshold value s₀ such that a random induced quantum state is entangled with very large probability if the environmental dimension s is smaller than s₀ and is not entangled with very large probability if s is bigger than s₀. An estimate for s₀ will be presented.