Local maximum property and q-plurisubharmonic functions in uniform algebras (Q1103823)

There is proven a formula expressing higher order Shilov boundaries of the tensor products of uniform algebras in terms of boundaries of the factor algebras. Main step: Cartesian product of k- and \(\ell\)-maximum sets is a \((k+\ell +1)\)-maximum set. (Definition: locally closed \(X\subset {\mathbb{C}}^ n\) is a k-maximum set if polynomials have local maximum property on intersections of X with (n-k)-dimensional complex planes.) Other properties and characterizations of k-maximum sets are given, e.g., \(X\subset {\mathbb{C}}^ n\) is a k-maximum set iff each k- plurisubharmonic function has local maximum property on X.