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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On some subalgebras of $ B(c\sb 0)$ and $ B(l\sb 1)$

Authors: F. P. Cass and J. X. Gao
Journal: Trans. Amer. Math. Soc. 347 (1995), 4461-4470
MSC: Primary 47D30; Secondary 46B25, 46B28
MathSciNet review: 1311904
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Abstract: For a non-reflexive Banach space $ X$ and $ w \in {X^{{\ast}{\ast}}}$, two families of subalgebras of $ B(X),\;{\Gamma _w} = \{ T \in B(X)\vert{T^{{\ast}{\ast}}}w = kw\;{\text{for some}}\;k \in \mathbb{C}{\text{\} }}$, and $ {\Omega _w} = \{ T \in B(X)\vert{T^{{\ast}{\ast}}}w \in w \oplus \hat X\} $ for $ w \in {X^{{\ast}{\ast}}}\backslash \hat X$ with $ {\Omega _w} = B(X)$ for $ w \in \hat X$, were defined originally by Wilansky. We consider $ X = {c_0}$ and $ X = {l_1}$ and investigate relationships between the subalgebras for different $ w \in {X^{{\ast}{\ast}}}$. We prove in the case of $ {c_0}$ that, for $ w \in {X^{{\ast}{\ast}}}\backslash \hat X$, all $ {\Gamma _w}$'s are isomorphic and all $ {\Omega _w}$ 's are isomorphic. For $ X = {l_1}$, where it is known that not all $ {\Gamma _w}$'s are isomorphic and not all $ {\Omega _w}$ 's are isomorphic, we show, surprisingly, that subalgebras associated with a Dirac measure on $ \beta \mathbb{N}\backslash \mathbb{N}$, regarded as a functional on $ l_1^{\ast}$, are isomorphic to those associated with some Banach limit (i.e., a translation invariant extended limit). We also obtain a representation for the operators in the subalgebras $ \{ \cap {\Gamma _f}\vert f\;{\text{is a Banach limit}}\} $ and $ \{ \cap {\Omega _f}\vert f\;{\text{is a Banach limit}}\} $ of $ B({l_1})$.

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Keywords: Subalgebras, operators on Banach spaces, algebraic isomorphism, Banach limits, Dirac measures, Stone-Čech compactification
Article copyright: © Copyright 1995 American Mathematical Society