Arens Regularity And Factorization Property (1007.3110v1)

Abstract: In this paper, we will study some Arens regularity properties of module actions. Let $B$ be a Banach $A-bimodule$ and let ${Z}^\ell_{B^{}}(A^{})$ and ${Z}^\ell_{A^{}}(B^{})$ be the topological centers of the left module action $\pi_\ell:~A\times B\rightarrow B$ and the right module action $\pi_r:~B\times A\rightarrow B$, respectively. In this paper, we will extend some problems from topological center of second dual of Banach algebra $A$, $Z_1(A^{**})$, into spaces ${Z}^\ell_{B^{}}(A^{})$ and ${Z}^\ell_{A^{}}(B^{})$. We investigate some relationships between ${Z}1({A^{**}})$ and topological centers of module actions. For an unital Banach $A-module$ $B$ we show that ${Z}^\ell{A^{}}(B^{}){Z}1({A^{**}})={Z}^\ell{A^{}}(B^{})$ and as results in group algebras, for locally compact group $G$, we have ${Z}^\ell_{{L^1(G)}^{}}(M(G)^{})M(G)={Z}^\ell_{{L^1(G)}^{}}(M(G)^{})$ and ${Z}^\ell_{M(G)^{}}({L^1(G)}^{})M(G)={Z}^\ell_{M(G)^{}}({L^1(G)}^{})$. For Banach $A-bimodule$ $B$, if we assume that $B^B^{*}\subseteq A^*$, then $~B^{}{Z}_1(A^{})\subseteq {Z}^\ell_{A^{}}(B^{})$ and moreover if $B$ is an unital as Banach $A-module$, then we conclude that $B^{}{Z}_1({A^{}})={Z}^\ell_{A^{}}(B^{})$. Let ${Z}^\ell_{A^{}}(B^{})A\subseteq B$ and suppose that $B$ is $WSC$, so we conclude that ${Z}^\ell_{A^{}}(B^{})=B$. If $\overline{B^{*}A}\neq B^*$ and $ B^{**}$ has a left unit $A^{**}-module$, then $Z^\ell_{B^{}}(A^{})\neq A^{**}$. We will also establish some relationships of Arens regularity of Banach algebras $A$, $B$ and Arens regularity of projective tensor product $A\hat{\otimes}B$.