A Dynamical Hierarchy of Banach Algebras

Abstract

Stability theory has caught the attention of mathematicians in many areas, such as model theory and functional analysis. In particular, in the early 80's, J.-L. Krivine and B. Maurey introduced the concept of stable Banach spaces. This stability has a significant impact on the geometry of such spaces. They proved that any separable infinite-dimensional stable Banach space contains a copy of l_p for some p∈[1,∞) almost isometrically. Recently, S. Ferri and M. Neufang introduced the notion of multiplicative stability of Banach algebras as an analogue of stability of Banach spaces in Krivine- Maurey's sense, to which they refer as additive stability. In this work, we investigate properties of multiplicative and additive stability of Banach algebras such as the lp-direct sum of a sequence of multiplicatively stable Banach algebras, and the relation between additive stability and Arens regularity in a certain class of Banach algebras. Further, we introduce hyper-instability as a strong version of multiplicative instability. Moreover, we study multiplicative stability of some well-known Banach algebras. We define and study a stronger and a weaker version of multiplicative stability, inspired by spaces of functions on topological semigroups, namely, almost periodic and tame functions. Based on our work, we introduce a dynamical hierarchy of Banach algebras, which is a new classification of Banach algebras. This classification puts dividing lines to measure, in a sense, multiplicative stability of Banach algebras.