Hyperreflexivity of the bounded n-cocycle spaces of Banach algebras

The concept of hyperreflexivity has previously been defined for subspaces of $B(X,Y)$, where $X$ and $Y$ are Banach spaces. We extend this concept to the subspaces of $B^n(X,Y)$, the space of bounded $n$-linear maps from $X\times\cdots\times X=X^{(n)}$ into $Y$, for any $n\in \mathbb{N}$. If $A$ is...

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Other Authors: Samei, Ebrahim
Language:English
Published: 2014
Subjects:
Online Access:http://hdl.handle.net/10388/ETD-2014-08-1653
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spelling ndltd-USASK-oai-ecommons.usask.ca-10388-ETD-2014-08-16532014-08-31T04:52:17ZHyperreflexivity of the bounded n-cocycle spaces of Banach algebrasBanach algebrasC^*-algebrasGroup algebrasBounded derivationsBounded n-cocyclesLocal unitsCSL and nest algebras.The concept of hyperreflexivity has previously been defined for subspaces of $B(X,Y)$, where $X$ and $Y$ are Banach spaces. We extend this concept to the subspaces of $B^n(X,Y)$, the space of bounded $n$-linear maps from $X\times\cdots\times X=X^{(n)}$ into $Y$, for any $n\in \mathbb{N}$. If $A$ is a Banach algebra and $X$ a Banach $A$-bimodule, we obtain sufficient conditions under which $\Zc^n(A,X)$, the space of all bounded $n$-cocycles from $A$ into $X$, is hyperreflexive. To do so, we define two notions related to a Banach algebra: The strong property $(\B)$ and bounded local units (b.l.u). We show that there are sufficiently many Banach algebras which have both properties. We will prove that all C$^*$-algebras and group algebras have the strong property $(\B).$ We also prove that finite CSL algebras and finite nest algebras have this property. We further show that for an arbitrary Banach algebra $A$ and each $n\geq 2$, $M_n(A)$ has the strong property $(\B)$ whenever it is equipped with a Banach algebra norm. In particular, this implies that all Banach algebras are embedded into a Banach algebra with the strong property $(\B)$. With regard to bounded local units, we show that all $C^*$-algebras and many group algebras have b.l.u. We investigate the hereditary properties of both notions to construct more example of Banach algebras with these properties. We apply our approach and show that the bounded $n$-cocycle spaces related to Banach algebras with the strong property $(\B)$ and b.l.u. are hyperreflexive provided that the space of the corresponding $n+1$-coboundaries are closed. This includes nuclear C$^*$-algebras, many group algebras, matrix spaces of certain Banach algebras and finite CSL and nest algebras. We finish the thesis with introducing {\it the hyperreflexivity constant}. We make our results more precise with finding an upper bound for the hyperreflexivity constant of the bounded $n$-cocycle spaces.Samei, Ebrahim2014-08-30T12:00:18Z2014-08-30T12:00:18Z2014-082014-08-29August 2014textthesishttp://hdl.handle.net/10388/ETD-2014-08-1653eng
collection NDLTD
language English
sources NDLTD
topic Banach algebras
C^*-algebras
Group algebras
Bounded derivations
Bounded n-cocycles
Local units
CSL and nest algebras.
spellingShingle Banach algebras
C^*-algebras
Group algebras
Bounded derivations
Bounded n-cocycles
Local units
CSL and nest algebras.
Hyperreflexivity of the bounded n-cocycle spaces of Banach algebras
description The concept of hyperreflexivity has previously been defined for subspaces of $B(X,Y)$, where $X$ and $Y$ are Banach spaces. We extend this concept to the subspaces of $B^n(X,Y)$, the space of bounded $n$-linear maps from $X\times\cdots\times X=X^{(n)}$ into $Y$, for any $n\in \mathbb{N}$. If $A$ is a Banach algebra and $X$ a Banach $A$-bimodule, we obtain sufficient conditions under which $\Zc^n(A,X)$, the space of all bounded $n$-cocycles from $A$ into $X$, is hyperreflexive. To do so, we define two notions related to a Banach algebra: The strong property $(\B)$ and bounded local units (b.l.u). We show that there are sufficiently many Banach algebras which have both properties. We will prove that all C$^*$-algebras and group algebras have the strong property $(\B).$ We also prove that finite CSL algebras and finite nest algebras have this property. We further show that for an arbitrary Banach algebra $A$ and each $n\geq 2$, $M_n(A)$ has the strong property $(\B)$ whenever it is equipped with a Banach algebra norm. In particular, this implies that all Banach algebras are embedded into a Banach algebra with the strong property $(\B)$. With regard to bounded local units, we show that all $C^*$-algebras and many group algebras have b.l.u. We investigate the hereditary properties of both notions to construct more example of Banach algebras with these properties. We apply our approach and show that the bounded $n$-cocycle spaces related to Banach algebras with the strong property $(\B)$ and b.l.u. are hyperreflexive provided that the space of the corresponding $n+1$-coboundaries are closed. This includes nuclear C$^*$-algebras, many group algebras, matrix spaces of certain Banach algebras and finite CSL and nest algebras. We finish the thesis with introducing {\it the hyperreflexivity constant}. We make our results more precise with finding an upper bound for the hyperreflexivity constant of the bounded $n$-cocycle spaces.
author2 Samei, Ebrahim
author_facet Samei, Ebrahim
title Hyperreflexivity of the bounded n-cocycle spaces of Banach algebras
title_short Hyperreflexivity of the bounded n-cocycle spaces of Banach algebras
title_full Hyperreflexivity of the bounded n-cocycle spaces of Banach algebras
title_fullStr Hyperreflexivity of the bounded n-cocycle spaces of Banach algebras
title_full_unstemmed Hyperreflexivity of the bounded n-cocycle spaces of Banach algebras
title_sort hyperreflexivity of the bounded n-cocycle spaces of banach algebras
publishDate 2014
url http://hdl.handle.net/10388/ETD-2014-08-1653
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