The concept of stability in the sense of Krivine–Maurey has proven very useful in Banach space geometry. We introduce and study a corresponding notion in the setting of Banach algebras, which we call multiplicative stability. As we shall see, the algebra of Schatten $p$-class operators on a separable Hilbert space is multiplicatively stable, where $p \in [1, \infty)$, while no infinite-dimensional $C^*$-algebra is. We also explore stronger and weaker versions of this concept, using the rich structure of spaces of functions defined on topological semigroups, including almost periodic, weakly almost periodic, and tame functions. This leads us to a novel classification of Banach algebras, providing a dynamical hierarchy. In this context, we also investigate further important classes of Banach algebras, such as group algebras and Fourier algebras, algebras of compact operators on Banach spaces, and algebras of differentiable functions.

The talk is based on joint work with (my former PhD student) Narjes Alabkary, (my former postdoctoral fellow) Reza Esmailvandi, and Stefano Ferri.