On some properties of \(A\) inherited by \(C_b(X,A)\). (English) Zbl 1414.46031

Greuel, Gert-Martin (ed.) et al., Singularities, algebraic geometry, commutative algebra, and related topics. Festschrift for Antonio Campillo on the occasion of his 65th birthday, University of Valladolid, Spain, June 19–23, 2017. Cham: Springer. 563-578 (2018).
Let \(X\) be a completely regular Hausdorff space, \(A\) a complete locally \(m\)-pseudoconvex algebra, the topology of which is given by a collection \(\{p_\alpha:\alpha\in\mathcal{A}\}\) of \(k_\alpha\)-homogeneous submultiplicative seminorms for \(k_\alpha\in(0,1]\) for each \(\alpha\in\mathcal{A}\). Moreover, let \(C_b(X,A)\) denote the algebra of all bounded and continuous \(A\)-valued maps and \(\{\tilde{p}_\alpha:\alpha\in\mathcal{A}\}\) the collection of \(k_\alpha\)-homogeneous seminorms on \(C_b(X,A)\), defined by \(\tilde{p}_\alpha (f)=\sup _{x\in X}p_\alpha(f(x))\) for each \(f\in C_b(X,A)\). Then \(C_b(X;A)\) is a complete locally \(m\)-pseudoconvex algebra. The main result in the present paper is: if \(A\) is a projective limit of \(k_\alpha\)-Banach algebras \(A_\alpha\), then \(C_b(X,A)\) is a projective limit of \(k_\alpha\)-Banach algebras \(C_b(X,A_\alpha)\). In addition, several properties from \(A\) are transformed to the topological algebra \(C_b(X,A)\).
Reviewers’ remark: (1) All seminorms of the form \(p_{k_\alpha}\) would better be written in the form \(p_\alpha\). To say that “\(p_{k_\alpha}\) is a \(k_\alpha\)-seminorm” is not correct, because \(k_\alpha\in(0,1]\) is a number, and \(\alpha\) is a member of the index set \(\Lambda\). (2) It is not clear why in Theorem 1 it was assumed that \(A\) is commutative.
For the entire collection see [Zbl 1401.14005].
Reviewer: Mati Abel (Tartu)


46H05 General theory of topological algebras
46E40 Spaces of vector- and operator-valued functions
46M40 Inductive and projective limits in functional analysis
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