A Dynamical Hierarchy of Banach Algebras

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.