We present several notions of joint spectral radius of mutually commuting elements of a locally convex algebra and prove that all of them yield the same value in case the algebra is pseudo-complete. This generalizes a result proved by the author in 1993 for elements of a Banach algebra.
2000 Mathematics Subject Classification: Primary 46H05, 46H20; Secondary 46M20.Every unital "combinatorially regular" commutative uniform complete locally m-convex algebra is local.
It is a survey talk concerning locally bounded algebras.
Let A be an algebra over the field of complex numbers with a (Hausdorff) topology given by a family Q = {qλ|λ ∈ Λ} of square preserving rλ-homogeneous seminorms (rλ ∈ (0, 1]). We shall show that (A, T(Q)) is a locally m-convex algebra. Furthermore we shall show that A is commutative.
Multi-dimensional generalizations of the Wiener-Żelazko and Lévy-Żelazko theorems are obtained.
We characterize elements in a semisimple Banach algebra which are quasinilpotent equivalent to maximal finite rank elements.
Let a real Banach algebra A with unit be ordered by an algebra cone K. We study the elements a ∈ A with exp(ta) ∈ K, t≥ 0.
We give a spectral characterisation of rank one elements and of the socle of a semisimple Banach algebra.
Without the "scarcity lemma", two kinds of "rank one elements" are identified in semisimple Banach algebras.