Translation invariant topologies on commutative \(^*\)-algebras
Let \(A\) be a commutative \(*\)-algebra with unit and \(P\) be the set of positive elements of \(A\). Under certain conditions on \(A\) the map \(a\to(a^*a)^{1/2}\) is a multiplicative \(*\)-preserving \(A\)-norm on \(A\). This norm induces an \(A\)-valued metric with an \(A\)-ball defined by \[ S(x,a)=\{y\in A:\;\| x-y\|\ll a\}\quad\text{for all }x\in A \] and strictly positive \(a\). This family of \(A\)-balls is a basis for a topology on \(A\) called the \(*\)-topology of \(A\) and denoted by \(\tau_ *\). It is shown that \((A,\tau_ *)\) is a Hausdorff \(Q\)-ring with continuous involution and inversion. Let \(C(X)\) be the algebra of continuous complex-valued functions on a topological space \(X\). It is shown that on \(C(X)\), \(\tau_ *\) is stronger than the compact-open topology and conditions for the equality of these topologies are given. The author also uses the results of \textit{A. Mallios} [J. Math. Anal. Appl. 92, 452-506 (1983; Zbl 0564.18011) and ibid. 106, 454-539 (1985; Zbl 0614.46064)] to derive further results about topological modules.
multiplicative \(*\)- preserving \(A\)-norm
Hausdorff \(Q\)-ring with continuous involution and inversion
scientific article; zbMATH DE number 60709