On some subalgebras of $B(c_ 0)$ and $B(l_ 1)$

Abstract:

For a non-reflexive Banach space $X$ and $w \in {X^{{\ast }{\ast }}}$, two families of subalgebras of $B(X),\;{\Gamma _w} = \{ T \in B(X)|{T^{{\ast }{\ast }}}w = kw\;{\text {for some}}\;k \in \mathbb {C}{\text {\} }}$, and ${\Omega _w} = \{ T \in B(X)|{T^{{\ast }{\ast }}}w \in w \oplus \hat X\}$ 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 $\{ \cap {\Gamma _f}|f\;{\text {is a Banach limit}}\}$ and $\{ \cap {\Omega _f}|f\;{\text {is a Banach limit}}\}$ of $B({l_1})$.