Schedule for: 22w2235 - Canadian Abstract Harmonic Analysis Symposium (CAHAS) 2020

Note: the Lecture rooms are available after 16:00.

Beverages and a small assortment of snacks are available in the lounge on a cash honour system.

A buffet breakfast is served daily between 7:00am and 9:00am in the Vistas Dining Room, the top floor of the Sally Borden Building. Note that BIRS does not pay for meals for 2-day workshops.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

We demonstrate constructions of Lévy processes, which we interpret as symmetric convolution semigroups of states, on locally compact quantum groups. We focus on quantum groups arising as Rieffel deformations of groups. Time permitting, we will explain why such objects are of interest. Based on joint work with Adam Skalski.

An $L^p$-operator algebra is a Banach algebra that admits an isometric representation on some $L^p$-space ($p \neq 2$). Given such an algebra $A$, we show that it contains a unique maximal sub-$C^\ast$-algebra, which we call its $C^\ast$-core. The $C^\ast$-core is automatically abelian, and its spectrum is naturally equipped with an inverse semigroup of partial homeomorphisms. We call the associated groupoid of germs the Weyl groupoid of $A$.

The Weyl groupoid contains information about the internal dynamics of the algebra $A$, and in some some cases it is a complete invariant. For example, given a topologically free action on a compact space, the Weyl groupoid of the reduced $L^p$-crossed product is simply the transformation groupoid of the action. This leads to strong rigidity results for reduced groupoid algebras and reduced crossed products.

We use our results to answer a question of Phillips: The $L^p$-analog of the Cuntz algebra $\mathcal{O}_2$ is not isomorphic to its tensor square.

(Joint work with Yemon Choi and Eusebio Gardella.)

In 1994 Johnson obtained an explicit formula for the amenability constant of the Fourier algebra of a finite group, from which it follows that for any non-abelian finite $G$, $\mathrm{AM}(A(G))$ is at least $3/2$. Subsequently, Forrest and Runde showed that the Fourier algebra of a locally compact group $G$ is amenable if and only if $G$ is virtually abelian; it follows from their method of proof and old results on norms of Schur multipliers that when $G$ is non-abelian the amenability constant of $A(G))$ is at least $2/ \sqrt{3}$. In this talk we report on recent work which shows that the lower bound (for arbitrary $G$) can be improved to 3/2 to match Johnson's result, even though there are virtually abelian groups in which each finite subgroup is abelian. Our approach relies on an associated minorant $\mathrm{AD}(G_d)$ and an explicit formula for this minorant when $G$ is virtually abelian, using the operator-valued Fourier transform for such groups. We also characterize those $G$ which attain the minimal value of $3/2$; if time permits we will comment on some further results for finite groups.

I will talk about amenability constant gap results for Banach algebras associated with groups. For several familiar algebras associated to finite groups their amenability constants can be described with very nice formulas involving representation theory, which opens the door to the use of computational methods. I will discuss recent investigations of this nature on the central Fourier algebra, which on a finite group is just the class functions equipped with the Fourier algebra norm.

The group $G_2=\mathbb{R}^2\rtimes \mathrm{GL}_2(\mathbb{R} )$ of invertible affine transformations of the plane has, up to unitary equivalence, exactly one square-integrable representation. We will introduce a factorization of $G_2$ into a product of closed subgroups and use this to derive an explicit expression for the square-integrable representation. As a result, we obtain an analog of the continuous wavelet transform for functions of $2+1$ variables.

In this talk, we review the theory of translation- and shift-invariant subspaces of $L^2(\mathbb{R}^n)$, highlighting some recent developments and applications. We then look at the case where our shifts are given by the action of a crystallographic group; more precisely, we examine the natural representation on $L^2(\mathbb{R}^n)$ given by this action. We give an explicit decomposition of this representation into a direct integral of irreducibles, and deduce a characterization of the (crystal-)shift-invariant subspaces. We discuss the value of this characterization for building a crystallographic wavelet theory.

A buffet lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. Note that BIRS does not pay for meals for 2-day workshops.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

It is well-known that when the second dual $L^1(G)^{\ast\ast}$ of the group algebra $L^1(G)$ of a locally compact group $G$ is furnished with one of the Arens multiplications, the only elements $p \in L^1(G)^{\ast\ast}$ for which both multiplication operators $q\mapsto pq$ and $q\mapsto qp$ are continuous are the elements of $L^1(G)$. In short, $Z_t(L^1(G)^{\ast \ast})=L^1(G)$, the topological center of $L^1(G)^{\ast \ast}$ is $L^1(G)$, i.e., it is as small as it gets. One says in this case that $L^1(G)$ is strongly Arens irregular.

It is also known (at least, since Ülger's 2011 paper Characterizations of Riesz sets) that ideals of $L^1(G)$ can be even Arens regular, i.e., that the multiplication on their biduals can even be (separately) continuous. In this talk, we will discuss the Arens regularity properties of ideals of $L^1(G)$ with $G$ compact and Abelian and will show that all sorts of behaviour are possible and actually occur. We will see that there is a correlation between these properties and the size of the ideals, though this is not the whole story. On our way we will be stressing those aspects that can be replicated on a wide family of Banach algebras that include the algebras $L^1(G) $ for $G$ compact (not necessarily Abelian) or the Fourier algebra $A(G)$ with $G$ amenable and discrete.

We develop a new method, based on non-vanishing of second cohomology groups, for proving the failure of lifting properties for full $C^\ast$-algebras of countable groups with (relative) property (T). Consequently, we show the full $C^\ast$*-algebras of $\mathbb{Z}^2\rtimes \mathrm{SL}(2,\mathbb {Z})$ and $\mathrm{SL}(n,\mathbb{Z})$, for $n\geq 3$, do not have the local lifting property (LLP). We also prove that the full $C^\ast$-algebras of a large class of groups with property (T) do not have the lifting property (LP). This is based on joint work with A. Ioana and P. Spaas.

Suppose that $G$ is a topological group and $T(G) \subset C^b(G)$ is the linear subspace spanned by all coefficient functions of continuous, finite-dimensional, irreducible unitary representations of $G$. If $\varphi \colon T(G) \longrightarrow \mathbb{C}$ is any linear functional such that $\varphi (|h|^2) \ge 0$ for all $h \in T(G)$, then the Tannaka-Krein theorem states that $\varphi$ is positive, and hence has a continuous extension to $AP(G)=\overline{T(G)}$. This theorem was used by Tannaka to develop a duality theory for compact groups, and was used by Hewitt and Zuckerman to obtain a generalization of classical Kronecker approximation theorem.

In this talk, we discuss an extension of Tannaka-Krein theorem for functions $\varphi \colon T_{\mathcal P}(S) \longrightarrow A$, with $\varphi (|f|^2)\ge 0$, where $S$ is a topological semigroup, $A$ is a $C^\ast$-algebra, and $\mathcal P$ is a suitable collection of representations of $S$. Following Hewitt--Zuckerman, we give an application of this result to an approximation problem on the character space of topological semigroups. When applied to the semigroups of $(\mathbb{R} , +)$, this leads to the following sharpened form of Kronecker's theorem: Let $h_1,\ldots , h_n$ be real numbers such that $\{1,h_1,\ldots , h_n\}$ is linearly independent over $\mathbb{Q}$, let $b_1,\ldots , b_n$ be arbitrary real numbers, and $\epsilon >0$. If $S$ is any semigroup of $(\mathbb{R} , +)$ which contains a nonzero rational number, then there exists some $x\in S$ such that \[ |\exp (2\pi i b_j)-\exp (2\pi i xh_j)|<\epsilon \qquad (j=1,2,\ldots , n). \] The result gives concrete information on how $x$ can be chosen in Kronecker's theorem. In particular, $x$ can be chosen to be a multiple of any prime number.

In the phase space formulation of quantum mechanics, states are represented through Wigner/characteristic functions on the underlying phase space, and observables are parametrized by the Weyl representation. Primary examples include bosonic systems, multipartite qudit systems and angle-number systems. For these systems, phase space methods underlie important concepts and techniques, such as bosonic Gaussian states and channels, sharp uncertainty principles, and the stabilizer formalism of quantum error correction.

Mathematically, quantum kinematical systems with finitely many degrees of freedom are described by a locally compact abelian group $G$ and a cocycle. The cocycle induces a symplectic (i.e., phase space) structure on $G\times\widehat{G}$, which encodes the canonical commutation relations of the associated (projective) Weyl representation. Such abstract quantum kinematical systems have been studied from a variety of perspectives, including finite-dimensional approximations, uncertainty relations and generalized metaplectic operators. In this work, we continue this program by developing a formalism to study Gaussian states and channels for general quantum kinematical systems.

I will quickly review the phase space formulation for bosonic/qudit systems and discuss its generalization to abstract (2-regular) Weyl systems. I will then introduce Gaussian states and channels for abstract Weyl systems and discuss some of our main results, including (1) a complete characterization of Gaussian states for arbitrary Weyl systems, (2) a characterization of pure states with non-negative Wigner functions for totally disconnected systems, and (3) single letter formulae for the quantum capacities and minimum output entropies for arbitrary Gaussian channels over finite Weyl systems.

This is joint work with Cédric Bény, Hun Hee Lee, Sang Jun Park and Sang-Gyun Youn.

In the early 1950s, Duffin and Schaeffer introduced the concept of discrete frames for a Hilbert space $\mathcal{H}$. In pursuit of "non-harmonic Fourier series", they defined discrete frames as countable sequences $\{e_j\}_{j=1}^\infty$ in $\mathcal H$ together with real constants $0< A \leq B< \infty$ such that \[ A\|f\|_{\mathcal H}^2 \leq \sum_{j=1}^\infty|\langle f,e_j \rangle |^2 \leq B\|f\|_{\mathcal H}^2, \] for all $f\in \mathcal H$.

Most past work in constructing discrete frames in high-dimensional Euclidean domains has been limited to existence results, with useful concrete constructions proving difficult to find for $L^2(\mathbb R^n)$ when $n\geq 3$. I will outline a construction obtained by discretizing a continuous wavelet transform for $L^2(\mathbb R^{n^2})$ which holds for all $n$. I will describe the motivations for the construction which could generalize these techniques to other semisimple Lie groups

This is joint work with Mahya Ghandehari.

The study of the theory of hypergroups is one of the important directions of research in modern mathematics. Moment functions play a basic role in probability theory. Moment functions, exponential monomials and polynomials are the basic building blocks of spectral synthesis. A natural generalization can be defined on hypergroups which leads to the concept of generalized moment function sequences. All these functions can be characterized by well-known functional equations. In this talk we describe generalized moment function sequences on hypergroup joins.

A subset of a group $\Gamma$ is a set of synthesis when it is the hull of exactly one closed ideal in the Fourier algebra of $\Gamma$. It is a result of Kaniuth and Lau that characterizes the groups where every subset is a set synthesis, and hence the closed ideals of their Fourier algebras is fully classified in terms of their hulls. In this talk we generalize the notion of synthesis to compact quantum groups and prove an analogue of Kaniuth and Lau's result for the closed left ideals of the quantum group algebras of compact quantum groups. We will also highlight some applications, which include a generalization of recent work of White to the quantum measure algebras of co-amenable compact quantum groups and when certain closed left ideals of quantum group algebras admit bounded right approximate identities, generalizing a special case of work due to Forrest and Forrest-Kaniuth-Lau-Spronk.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. Note that BIRS does not pay for meals for 2-day workshops.

In this talk we will focus on the quantum channels (i.e. completely positive trace preserving maps between matrix algebras) satisfying compact group symmetry, i.e. covariant channels. The main point is that we could provide a rather complete description of covariant channels when the symmetry satisfies the "multiplicity-free" condition. This leads us to an effective parametrization of those channels, which might be useful in answering some open questions in QIT. We will visit a few examples including the case of Weyl covariant channels using projective representations. If time permits, we will mention what we can do with quantum group symmetry.

The Fourier algebra $A(G)$ of a locally compact group $G$, introduced by Eymard, is one of the favourite objects in abstract harmonic analysis. It has an advantage to be commutative that allows one to examine its Gelfand spectrum, which is known to be topologically isomorphic to $G$; the fact makes a non-trivial connection between Banach algebras and groups. We will discuss a weighted variant of Fourier algebra and show its connection with complexification of the underlying group. For compact groups this was done thanks to abstract complexification due to McKennon [Crelle, 79'] and Cartwright/McMullen [Crelle, 82']; the complexification is identified as the set of solutions to the equation $\Gamma(X)=X\otimes X$ in the set of elements affiliated with the von Neumann algebra $VN(G)$, here $\Gamma$ is the comultiplication on $VN(G)$. We extended this theory to general locally compact groups and use the model to describe the Gelfand spectrum of weighted Fourier algebras, showing that the latter is a part of the complexification for a wide class of locally compact groups and weights. I shall also present different examples of weights and determine the spectrum of the corresponding algebras. %I shall discuss Beurling--Fourier (or "weighted" Fourier) algebras $A(G,W)$ on various Lie groups $G$ focusing on their spectral analysis. If $G$ is a compact group and the weight $W$ is a so-called central weight given by a function $W:\widehat G\to\mathbb R^+$ on the dual $\widehat G$ of $G$, then the spectrum of $A(G,W)$ is a subset of the complexification of $G$ which was shown in [J.Ludwig, N.Spronk and L.Turowska, J. Func. Anal. (2012)]. This was possible thanks to the abstract Lie theory developed by McKennon and Cartwright/McMullen. Unfortunately for non-compact groups the model of abstract Lie theory is not compatible with our model of Beurling-Fourier algebra and the situation forces us to build our own connection between spectrum of $A(G,W)$ and the complexification of $G$. In this talk I will discuss different examples of weights, possibly non-central ones, and spectrum of the corresponding Beurling-Fourier algebras emphasizing their connection with the complexification of the underlying groups.

This talk is based on a joint work with Olof Giselsson.

For a locally compact abelian group $G$, J. L. Taylor (1971) gave a complete characterization of invertible elements in the measure algebra $M(G)$. Using the Fourier-Stieltjes transform, this characterization can be carried out in the context of Fourier-Stieltjes algebras $B(G)$. We generalise this characterization to the setting of the Fourier-Stieltjes algebra $B(G)$ of certain classes of locally compact groups, in particular, many totally minimal groups and the $ax+b$-group.

I will give a brief overview of amenability theory in the sense of B. Johnson as it applies to operator algebras, and indicate how it effects certain algebras of harmonic analysis. I will highlight recent work, partially of mine, around group $C^\ast$-algebras, and Fourier--Stieltjes algebras.

Let $G$ be a locally compact abelian group, let $m \in L^\infty(\hat{G} \times \hat{G})$, and let $\Phi_1 , \Phi_2$, $\Phi_3$ be Young functions. We consider the Orlicz spaces $L^{\Phi_1}(G)$, $L^{\Phi_2}(G)$ and $L^{\Phi_3}(G)$, and investigate the boundedness of the bilinear multiplier operator $B_m$ corresponding to a multiplier $m$. If $B_m$ defines a bounded bilinear operator from $L^{\Phi_1}(G) \times L^{\Phi_2}(G)$ to $L^{\Phi_3}(G)$, then $m$ is called a bilinear multiplier of type $(\Phi_1,\Phi_2,\Phi_3)$. We extend some existing results in $\mathbb{R}^n$ and give also examples of bilinear multipliers of type $(\Phi_1,\Phi_2,\Phi_3)$ in the setting of locally compact abelian groups.

This presentation is based on joint work with Alen Osançlıol.