It is shown, using geometric methods, that there is a C*-algebraic quantum group related to any double Lie group (also known as a matched pair of Lie groups or a bicrossproduct Lie group). An algebra underlying this quantum group is the algebra of a differential groupoid naturally associated with the double Lie group.

We tackle R. V. Kadison’s similarity problem (i.e. any bounded representation of any unital C*-algebra is similar to a *-representation), paying attention to the class of C*-unitarisable groups (those groups G for which the full group C*-algebra C*(G) satisfies Kadison’s problem) and thereby we establish some stability results for Kadison’s problem. Namely, we prove that $A{\otimes }_{min}B$ inherits the similarity problem from those of the C*-algebras A and B, provided B is also nuclear. Then we prove that G/Γ is...

Let G be a non-discrete locally compact group, A(G) the Fourier algebra of G, VN(G) the von Neumann algebra generated by the left regular representation of G which is identified with A(G)*, and WAP(Ĝ) the space of all weakly almost periodic functionals on A(G). We show that there exists a directed family ℋ of open subgroups of G such that: (1) for each H ∈ ℋ, A(H) is extremely non-Arens regular; (2) $VN\left(G\right)={\bigcup }_{H\in \mathscr{H}}VN\left(H\right)$ and $VN\left(G\right)/WAP\left(Ĝ\right)={\bigcup }_{H\in \mathscr{H}}[VN\left(H\right)/WAP\left(Ĥ\right)]$; (3) $A\left(G\right)={\bigcup }_{H\in \mathscr{H}}A\left(H\right)$ and it is a WAP-strong inductive union in the sense that the unions in (2) are strongly...

Generalizing the notion of matched pair of groups, we define and study matched pairs of locally compact groupoids endowed with Haar systems, in order to give new examples of measured quantum groupoids.

Let $C{V}_p\left(F\right)$ be the left convolution operators on $L^p\left(G\right)$ with support included in F and $M_p\left(F\right)$ denote those which are norm limits of convolution by bounded measures in M(F). Conditions on F are given which insure that $C{V}_p\left(F\right)$, $C{V}_p\left(F\right)/M_p\left(F\right)$ and $C{V}_p\left(F\right)/W$ are as big as they can be, namely have $l^{\infty }$ as a quotient, where the ergodic space W contains, and at times is very big relative to $M_p\left(F\right)$. Other subspaces of $C{V}_p\left(F\right)$ are considered. These improve results of Cowling and Fournier, Price and Edwards, Lust-Piquard, and others.

Let $\tau$ be a continuous unitary representation of the locally compact group $G$ on the Hilbert space $H_{\tau }$. Let $C_{\tau }^{*}[V{N}_{\tau }]$ be the $C^{*}[W^{*}]$ algebra generated by$$(L^1\left(G\right))\text{and}{M}_{\tau }(C_{\tau }^{*})=\left\{\phi \in V{N}_{\tau };\phi C_{\tau }^{*}+C_{\tau }^{*}\phi \subset C_{\tau }^{*}\right\}.$$The main result obtained in this paper is Theorem 1:If $G$ is $\sigma$-compact and $M_{\tau }(C_{\tau }^{*})=V{N}_{\tau }$ then supp $\tau$ is discrete and each $\pi$ in supp $\tau$ in CCR.We apply this theorem to the quasiregular representation $\tau ={\pi }_H$ and obtain among other results that $M_{{\pi }_H}(C_{{\pi }_H}^{*})=V{N}_{{\pi }_H}$ implies in many cases that $G/H$ is a compact coset space.