Hyperreflexivity of the bounded $n$-cocycle spaces of Banach algebras with matrix representations
Tom 250 / 2020
Studia Mathematica 250 (2020), 35-55
MSC: Primary 47B47.
DOI: 10.4064/sm180611-5-11
Opublikowany online: 29 July 2019
Streszczenie
We study the hyperreflexivity of the bounded $n$-cocycle spaces of Banach algebras with matrix representations. We show how representing elements of a Banach algebra as matrices helps us to prove that the Banach algebra has the strong property $(\mathbb {B})$ with a constant. Consequently, we can prove that if some conditions on the Hochschild cohomology groups are satisfied, then the spaces of bounded $n$-cocycles related to this type of Banach algebras are hyperreflexive and we can provide a bound for their hyperreflexivity constant. This approach in particular can be applied to matrix spaces of arbitrary Banach algebras, finite nest algebras on arbitrary Hilbert spaces and finite CSL algebras on separable Hilbert spaces.
