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Module homomorphisms and topological centres associated with weakly sequentially complete Banach algebras. (English) Zbl 0911.46030
Let $$A$$ be a complex Banach algebra with second dual $$A^{**}$$, also a Banach algebra. The topological centre $$\Lambda(A^{**})$$ is the set of elements in $$A^{**}$$ yielding a weak$$^*$$-continuous right multiplication. The authors begin with a review of results of M. Grosser [“Bidualräume und Vervollständigungen von Banachmodulen”, Lect. Notes Math. 717 (1979; Zbl 0412.46005)] on the structure of $$A^{**}$$ and $$A^*A$$, in particular that the existence of a bounded right approximate identity in $$A$$ implies that $$A^{**}$$ is isomorphic in the category of Banach spaces and continuous linear maps to $$(A^*A)^*\oplus (A^*A)^{\perp}$$. They then give criteria for a weakly sequential complete Banach algebra $$A$$ that $$\Lambda(A^{**})= A$$ and that the only $$\lambda$$ in $$\Lambda(A^{**})$$ with $$\lambda A\subseteq A$$ are themselves in $$A$$.
The next main result provides that under suitable hypotheses every continuous linear transformation $$T$$ from $$A^*$$ to $$A^*A$$ with $$T(xf)= xTf$$ is implemented by a right multiplication by an element of $$A$$. Special cases are treated: group algebras, the Volterra algebra and $$\ell^1(S)$$ for a commutative discrete semigroup $$S$$.

##### MSC:
 46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) 46H05 General theory of topological algebras
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