Modules over operator algebras and its module maps (Q1369005)

For Banach algebras \(A\) and \(B\) and a Banach \(A\)-\(B\)-bimodule let \(M(X)=\{m: A\times B\rightarrow X|\) \(m(ab,cd)=am(b,c)d\) for all \(a,b\in A\), \(c,d\in B\}\) and \(\phi: X\rightarrow M(X)\) be defined by \(\phi(x)(a,b)=axb\) for all \(a\in A\), \(b\in B\), and \(x\in X\). Let \(\tau\) denote the topology of the norm \(\| m\|= \sup\{\| m(a,b)\|: a\in A\), \(b\in B\), \(\| a \|\leq 1\), \(\| b\|\leq 1\}\) on \(M(X)\) and \(\mu\) the topology generated by a family of seminorms \(p_{a,b}(m)=\| m(a,b)\|\). The author investigates properties of the space \((M(X),\mu)\). The main result contained in the paper is the following: Theorem. Let \(A\) and \(B\) be \(C^\ast\)-algebras and \(X\) be a Banach \(A\)-\(B\)-bimodule. Let \((e_\alpha)\), \((e_\beta')\) be minimal approximate identities of \(A\) and \(B\) respectively. A subset \(N\) of \((M(X),\mu)^\ast\) is equicontinuous if and only if the following two conditions are satisfied: \(\bullet\) \(N\) is norm bounded in \((M(X),\tau)^\ast\); \(\bullet\) \(\| f-e_\alpha\cdot f\cdot e'_\beta\|\rightarrow 0\) uniformly for \(f\in N\), where \((a\cdot f\cdot b)(m)=f(\phi(m(a,b))\).