  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
$A\hat{\otimes}B$.
