Related papers: Uniform continuity over locally compact quantum gr…

The following topics are presented in these notes: Elements of Banach algebras, Banach algebras of the form $L^1(G)$, where $G$ is a locally compact group, spectrum of elements of Banach algebras, the spectral theory of compact operators on…

With every locally compact group $G$, one can associate several interesting bi-invariant subspaces $X(G)$ of the weakly almost periodic functions $\mathrm{WAP}(G)$ on $G$, each of which captures parts of the representation theory of $G$.…

Let $G$ be a locally compact group. In this paper, we study various invariant subspaces of the duals of the algebras $A_M(G)$ and $A_{cb}(G)$ obtained by taking the closure of the Fourier algebra $A(G)$ in the multiplier algebra $MA(G)$ and…

In this paper, we quantize universal gauge groups such as SU(\infty), as well as their homogeneous spaces, in the sigma-C*-algebra setting. More precisely, we propose concise definitions of sigma-C*-quantum groups and sigma-C*-quantum…

We establish the dual equivalence of the category of (potentially nonunital) operator systems and the category of pointed compact nc (noncommutative) convex sets, extending a result of Davidson and the first author. We then apply this dual…

In this paper, we collect some technical results about weights on C*-algebras which are useful in de theory of locally compact quantum groups in the C*-algebra framework. We discuss the extension of a lower semi-continuous weight to a…

In 1972, B. E. Johnson proved that a locally compact group $G$ is amenable if and only if certain Hochschild cohomology groups of its convolution algebra $L^1(G)$ vanish. Similarly, $G$ is compact if and only if $L^1(G)$ is biprojective: In…

Let $\ell$ be a length function on a group $G$, and let $M_{\ell}$ denote the operator of pointwise multiplication by $\ell$ on $\bell^2(G)$. Following Connes, $M_{\ell}$ can be used as a ``Dirac'' operator for $C_r^*(G)$. It defines a…

We prove that if a connected and simply connected Lie group $G$ admits connected closed normal subgroups $G_1\subseteq G_2\subseteq \cdots \subseteq G_m=G$ with $\dim G_j=j$ for $j=1,\dots,m$, then its group $C^*$-algebra has closed…

Let $G$ be a locally compact abelian Hausdorff topological group which is non-compact and whose Pontryagin dual $\Gamma$ is partially ordered. Let $\Gamma^{+}\subset\Gamma$ be the semigroup of positive elements in $\Gamma$. The Hardy space…

We prove that the $L^1$-algebra of any non-Kac type compact quantum group does not satisfy operator biflatness. Since operator amenability implies operator biflatness, this result shows that any co-amenable, non-Kac type compact quantum…

We prove a number of results having to do with equipping type-I $\mathrm{C}^*$-algebras with compact quantum group structures, the two main ones being that such a compact quantum group is necessarily co-amenable, and that if the…

We devise a fairly general method for estimating the size of quotients between algebras of functions on a locally compact group. This method is based on the concept of interpolation sets and unifies the approaches followed by many authors…

Let G be a locally compact group, A(G) its Fourier algebra and L1(G) the space of Haar integrable functions on G. We study the Segal algebra SA(G)=A(G)\cap L1(G) in A(G). It admits an operator space structure which makes it a completely…

We study the structure of C*-algebras associated with compactly aligned product systems over group embeddable right LCM-semigroups. Towards this end we employ controlled maps and a controlled elimination method that associates the original…

We construct some inverse-closed algebras of bounded integral operators with operator-valued kernels, acting in spaces of vector-valued functions on locally compact groups. To this end we make systematic use of covariance algebras…

Let $S$ be a right reversible semitopological semigroup, and let $\operatorname{LUC}(S)$ be the space of left uniformly continuous functions on $S$. Suppose that $\operatorname{LUC}(S)$ has a left invariant mean. Let $K$ be a weakly compact…

We propose a definition of a "$C^*$-Eberlein" algebra, which is a weak form of a $C^*$-bialgebra with a sort of "unitary generator". Our definition is motivated to ensure that commutative examples arise exactly from semigroups of…

Let $G$ be a second countable locally compact groupoid equipped with a Haar system $\lambda$.In this work, we introduce and develop the notion of amenability for continuous unitary representations of $G$, formulated in terms of Hilbert…

This work provides a generalization of the Gelfand duality to the context of noncommutative locally $C^*$ algebras. Using a reformulation of a theorem proven by Dauns and Hofmann in the 60's we show that every locally $C^*$ algebra can be…