HYPERREFLEXIVITY CONSTANTS OF THE BOUNDED -COCYCLE SPACES OF GROUP ALGEBRAS AND C*-ALGEBRAS

We introduce the concept of strong property $(\mathbb{B})$ with a constant for Banach algebras and, by applying a certain analysis on the Fourier algebra of the unit circle, we show that all C*-algebras and group algebras have the strong property $(\mathbb{B})$ with a constant given by $288\unicode[STIX]{x1D70B}(1+\sqrt{2})$ . We then use this result to find a concrete upper bound for the hyperreflexivity constant of ${\mathcal{Z}}^{n}(A,X)$ , the space of bounded $n$ -cocycles from $A$ into $X$ , where $A$ is a C*-algebra or the group algebra of a group with an open subgroup of polynomial growth and $X$ is a Banach $A$ -bimodule for which ${\mathcal{H}}^{n+1}(A,X)$ is a Banach space. As another application, we show that for a locally compact amenable group $G$ and $1<p<\infty$ , the space $CV_{P}(G)$ of convolution operators on $L^{p}(G)$ is hyperreflexive with a constant given by $384\unicode[STIX]{x1D70B}^{2}(1+\sqrt{2})$ . This is the generalization of a well-known result of Christensen [‘Extensions of derivations. II’, Math. Scand. 50 (1) (1982), 111–122] for $p=2$ .

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