Concerning locally convex algebras without generalized topological divisors of zeros (Q910672)

The author generalizes to a larger class of topological algebras the Shilov theorem asserting that a Banach algebra either has topological divisors of zero or is isomorphic to the algebra of complex numbers. It is shown that if A is a complex LMC-algebra (i.e., a locally convex algebra such that, for each seminorm \(\| \|_{\alpha}\) defining the topology, one has \(\| xy\|_{\alpha}\leq \| x\|_{\alpha}\| y\|_{\alpha})\), then, if \(A={\mathbb{C}}\), A admits generalized topological divisors of zero (i.e., there exist two nonempty subsets \(S_ 1\), \(S_ 1\) of A such that \(0\not\in \bar S_ 1\cup \bar S_ 2\) and \(0\in \bar S_ 1\cdot \bar S_ 2)\).