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

Part of: Abstract harmonic analysis Special classes of linear operators Selfadjoint operator algebras

Published online by Cambridge University Press:  30 April 2019

EBRAHIM SAMEI  and
JAFAR SOLTANI FARSANI
EBRAHIM SAMEI*
Affiliation:
Department of Mathematics & Statistics, University of Saskatchewan, 106 Wiggins Road, Saskatoon, SK, Canada email samei@math.usask.ca Institute of Mathematics of the Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warszawa, Poland
JAFAR SOLTANI FARSANI
Affiliation:
Department of Mathematics & Statistics, University of Saskatchewan, 106 Wiggins Road, Saskatoon, SK, Canada email jafar.solt@gmail.com
*
✉samei@math.usask.ca
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Abstract

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 $p=2$.

Keywords

reflexivityhyperreflexivityhyperreflexivity constantn-cocyclesC*-algebrasgroup algebrasgroups with polynomial growthamenability

MSC classification

Primary: 47B47: Commutators, derivations, elementary operators, etc.
Secondary: 46L05: General theory of $C^*$-algebras 43A20: $L^1$-algebras on groups, semigroups, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 109 , Issue 1 , August 2020 , pp. 112 - 130
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Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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