On the continuity of intertwining operators over generalized convolution algebras (Q6626998)

Let \(\mathfrak{B}\) be a Banach algebra, \(\mathcal{X}_1\) a Banach \(\mathfrak{B}\)-bimodule and \(\mathcal{X}_2\) a weak Banach \(\mathfrak{B}\)-bimodule. Let also \(\theta:\mathcal{X}_1\rightarrow\mathcal{X}_2\) be a \(\mathfrak{B}\)-intertwining operator, that is, for every \(b\in\mathfrak{B}\), the maps \N\[\N\xi\mapsto\theta(b\xi)-b\theta(\xi)\quad\hbox{and}\quad \xi\mapsto\theta(b\xi)-\theta(\xi)b \N\]\Nfrom \(\mathcal{X}_1\) into \(\mathcal{X}_2\) are continuous.\N\NIn this paper, the author studies the automatic continuity of \(\theta\). He proves the following results under certain conditions.\N\N1. If \(\mathfrak{B}\) is either an \(A^*\)-algeabra or \(L^1(G|\mathscr{C})\), the algebra of integrable cross-sections, and \(\mathcal{X}_1=\mathfrak{B}\), then \(\theta\) is continuous if and only if \(\mathscr{L}(\theta)\) is closed, where \N\[\N\mathscr{L}(\theta)=\{b\in \mathfrak{B}|\; b\mathscr{S}(\theta)=\mathscr{S}(\theta)b=\{0\}\}, \N\]\Nin which \N\[\N\mathscr{S}(\theta)=\{\eta\in\mathcal{X}_2|\; \exists \{\xi_n\}_{n\in\mathbb{N}}\subseteq\mathcal{X}_1\text{ such that }\xi_n\rightarrow 0\text{ and }\theta(\xi_n)\rightarrow\eta\}. \N\]\NIn the case where \(\mathfrak{B}=L^1(G|\mathscr{C})\) if \(\mathcal{X}_2\) is a Banach \(\mathfrak{B}\)-bimodule, then \(\theta\) is continuous.\N\N2. If \(\mathfrak{B}\) is either \(\ell^1(G|\mathscr{C})\) or \(\ell^1_\alpha(G, C(X))\), the convolution algebra associated with the action \(\alpha\), then \(\theta\) is continuous.