On the fundamental inequality in locally multiplicatively convex algebras. (English) Zbl 0547.46029

In this paper a characterisation of the complete Hermitian locally multiplicatively convex (LMC) algebras is given. This is done by a set of inequalities concerning spectra which corresponds to the V. Pták’s fundamental inequality for the case of the algebra being a Banach algebra. It is proved that the complete LMC star algebra A is Hermitian if and only if for each index \(\alpha\) from a suitable set \(\Sigma\) the following inequality is fullfilled: \(| x|^{\alpha}_{\sigma}\leq (| x^*x|^{\alpha}_{\sigma})^{{1\over2}}\) where \(x\in A\) is arbitrary and \(\{||^{\alpha}_{\sigma}\}_{\alpha\in \Sigma}\) is the family of spectral radii corresponding to the family of seminorms \(\{q_{\alpha}\}_{\alpha\in \Sigma}\) which generates the topology in A.


46H05 General theory of topological algebras
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