On an expression of the spectrum in \(BP^ *\)-algebras (Q1105814)

An example is given of a commutative complete locally A-convex algebra A with an identity (the locally convex topology of A is given by a family of seminorms \(P_{\lambda}\) such that for every \(x\in A\) there exist numbers M(x) and N(x) such that for each \(P_{\lambda}:\) \(P_{\lambda}(xy)\leq M(x)P_{\lambda}(y)\) and \(P_{\lambda}(yx)\leq N(x)P_{\lambda}(y)\) for each \(y\in A)\) in which the spectrum \(\sigma\) (x) is strictly greater than the range of \(\kappa\) (x), as \(\kappa\) ranges over all continuous multiplicative linear functionals. To obtain the whole spectrum it is necessary to use all multiplicative linear functionals. This contradicts a claim made in [\textit{T. Husain}, \textit{S. A. Warsi}, ibid. 8, 15-28 (1977; Zbl 0329.46056)].