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Hyperreflexivity of the bounded n-cocycle spaces of Banach algebras

dc.contributor.advisorSamei, Ebrahimen_US
dc.contributor.committeeMemberWang, Jiun-Chau (JC)en_US
dc.contributor.committeeMemberTymchatyn, Eden_US
dc.contributor.committeeMemberBickis, Miken_US
dc.contributor.committeeMemberDick, Raineren_US
dc.creatorSoltani Farsani, Jafaren_US 2014en_US
dc.description.abstractThe 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.en_US
dc.subjectBanach algebrasen_US
dc.subjectGroup algebrasen_US
dc.subjectBounded derivationsen_US
dc.subjectBounded n-cocyclesen_US
dc.subjectLocal unitsen_US
dc.subjectCSL and nest algebras.en_US
dc.titleHyperreflexivity of the bounded n-cocycle spaces of Banach algebrasen_US
dc.type.materialtexten_US and Statisticsen_US of Saskatchewanen_US of Philosophy (Ph.D.)en_US


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