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

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 $A_\pi\subseteq B_\pi$)

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

Weak amenability of weighted Orlicz algebras

Let G be a locally compact abelian group, $\omega:G\to (0,\infty)$ be a weight, and ($\Phi$,$\Psi$) be a complementary pair of strictly increasing continuous Young functions. We show that for the weighted Orlicz algebra $L^\Phi_\omega(G)$, the weak amenability is obtained under conditions similar to the one considered by Y. Zhang for weighted group algebras. Our methods can be applied to various families of weighted Orlicz algebras, including weighted $L^p$-spaces

Extension of derivations, and Connes-amenability of the enveloping dual Banach algebra

If $D:A \to X$ is a derivation from a Banach algebra to a contractive, Banach $A$-bimodule, then one can equip $X^{**}$ with an $A^{**}$-bimodule structure, such that the second transpose $D^{**}: A^{**} \to X^{**}$ is again a derivation. We prove an analogous extension result, where $A^{**}$ is replaced by \F(A), the \emph{enveloping dual Banach algebra} of $A$, and $X^{**}$ by an appropriate kind of universal, enveloping, normal dual bimodule of $X$. Using this, we obtain some new characterizations of Connes-amenability of \F(A). In particular we show that \F(A) is Connes-amenable if and only if $A$ admits a so-called WAP-virtual diagonal. We show that when $A=L^1(G)$, existence of a WAP-virtual diagonal is equivalent to the existence of a virtual diagonal in the usual sense. Our approach does not involve invariant means for $G$.Comment: v2: AMS-LaTeX, 11pt, 40 pages. Various minor improvements and corrections, including changes to notation and additional references; also new material in Sections 5 and 6. Incorporates referee's revisions. To appear in Mathematica Scandinavic