Weak amenability and 2-weak amenability of Beurling algebras

Let L^1_\om(G) be a Beurling algebra on a locally compact abelian group $G$. We look for general conditions on the weight which allows the vanishing of continuous derivations of L^1_\om(G). This leads us to introducing vector-valued Beurling algebras and considering the translation of operators on them. This is then used to connect the augmentation ideal to the behavior of derivation space. We apply these results to give examples of various classes of Beurling algebras which are weakly amenable, 2-weakly amenable or fail to be even 2-weakly amenable.Comment: 25 page

On local properties of Hochschild cohomology of a C∗^*- algebra

Let $A$ be a C$^*$-algebra, and let $X$ be a Banach $A$-bimodule. B. E. Johnson showed that local derivations from $A$ into $X$ are derivations. We extend this concept of locality to the higher cohomology of a $C^*$-algebra %for $n$-cocycles from $A^{(n)}$ into $X$ and show that, for every $n\in \N$, bounded local $n$-cocycles from $A^{(n)}$ into $X$ are $n$-cocycles.Comment: 13 page

Exotic C*-algebras of geometric groups

We consider a new class of potentially exotic group C*-algebras $C^*_{PF_p^*}(G)$ for a locally compact group $G$, and its connection with the class of potentially exotic group C*-algebras $C^*_{L^p}(G)$ introduced by Brown and Guentner. Surprisingly, these two classes of C*-algebras are intimately related. By exploiting this connection, we show $C^*_{L^p}(G)=C^*_{PF_p^*}(G)$ for $p\in (2,\infty)$, and the C*-algebras $C^*_{L^p}(G)$ are pairwise distinct for $p\in (2,\infty)$ when $G$ belongs to a large class of nonamenable groups possessing the Haagerup property and either the rapid decay property or Kunze-Stein phenomenon by characterizing the positive definite functions that extend to positive linear functionals of $C^*_{L^p}(G)$ and $C^*_{PF_p^*}(G)$. This greatly generalizes earlier results of Okayasu and the second author on the pairwise distinctness of $C^*_{L^p}(G)$ for $2<p<\infty$ when $G$ is either a noncommutative free group or the group $SL(2,\mathbb R)$, respectively. As a byproduct of our techniques, we present two applications to the theory of unitary representations of a locally compact group $G$. Firstly, we give a short proof of the well-known Cowling-Haagerup-Howe Theorem which presents sufficient condition implying the weak containment of a cyclic unitary representation of $G$ in the left regular representation of $G$. Also we give a near solution to a 1978 conjecture of Cowling. This conjecture of Cowling states if $G$ is a Kunze-Stein group and $\pi$ is a unitary representation of $G$ with cyclic vector $\xi$ such that the map $$G\ni s\mapsto \langle \pi(s)\xi,\xi\rangle$$ belongs to $L^p(G)$ for some $2< p <\infty$, then $A_\pi\subseteq L^p(G)$. We show $B_\pi\subseteq L^{p+\epsilon}(G)$ for every $\epsilon>0$ (recall $A_\pi\subseteq B_\pi$)

Quotients of Fourier algebras, and representations which are not completely bounded

We observe that for a large class of non-amenable groups $G$, one can find bounded representations of $A(G)$ on Hilbert space which are not completely bounded. We also consider restriction algebras obtained from $A(G)$, equipped with the natural operator space structure, and ask whether such algebras can be completely isomorphic to operator algebras; partial results are obtained, using a modified notion of Helson set which takes account of operator space structure. In particular, we show that if $G$ is virtually abelian, then the restriction algebra $A_G(E)$ is completely isomorphic to an operator algebra if and only if $E$ is finite.Comment: v3: 10 pages, minor edits and slight change to title from v2. Final version, to appear in Proc. Amer. Math. So

Weighted discrete hypergroups

Weighted group algebras have been studied extensively in Abstract Harmonic Analysis where complete characterizations have been found for some important properties of weighted group algebras, namely amenability and Arens regularity. One of the generalizations of weighted group algebras is weighted hypergroup algebras. Defining weighted hypergroups, analogous to weighted groups, we study Arens regularity and isomorphism to operator algebras for them. We also examine our results on three classes of discrete weighted hypergroups constructed by conjugacy classes of FC groups, the dual space of compact groups, and hypergroup structure defined by orthogonal polynomials. We observe some unexpected examples regarding Arens regularity and operator isomorphisms of weighted hypergroup algebras.Comment: 27 pages. This version is shorter but still covers all the main results of the previous on