‎let $x$‎, ‎$y$ and $z$ be banach spaces and $f:xtimes y‎ ‎longrightarrow z$ a bounded bilinear map‎. ‎in this paper we‎ ‎study the relation between arens regularity of $f$ and the‎ ‎reflexivity of $y$‎. ‎we also give some conditions under which the‎ ‎arens regularity of a banach algebra $a$ implies the arens‎ ‎regularity of certain banach right module action of $a$‎ .

In a previous paper (Arens; these references are to the bibliography of the present paper) there was investigated (in a more general, abstract setting) the process of forming the adjoint operation m*\ Z-XX^Ydefined, for fEZ~, xEX, by TM*(f, x)(y) = f(m(x, y)) (y £ F). The simple proof that m* satisfies 1.1, 1.2, and 1.3 with the same value of M is given in (Arens). This construction can be iter...

‎We present a characterization of Arens regular semigroup algebras‎ ‎$ell^1(S)$‎, ‎for a large class of semigroups‎. ‎Mainly‎, ‎we show that‎ ‎if the set of idempotents of an inverse semigroup $S$ is finite‎, ‎then $ell^1(S)$ is Arens regular if and only if $S$ is finite‎.