The definition of amenable Banach algebras was introduced following Johnson's theorem (1972), which provides an equivalent condition for the amenability of locally compact groups in terms of convolution Banach algebras and derivations. I will present a general overview of the development of research on the property of amenability, including its connections with the Banach--Tarski paradox, and then move on to results concerning amenability for vector-valued Banach algebras, namely the relationships between the amenability properties of a Banach algebra $A$ and those of the algebra $\ell_p(A)$, where $1 \leq p < \infty$. I will also present several facts concerning vector-valued Banach algebras and their amenability that are already known to us.

The above investigations led to a theorem (and the main result of joint work [Koczorowski and Piszczek, 2024]) that provides an equivalent characterization of weakly amenable Banach algebras $A$ in terms of the existence of a certain constant. This introduces the definition of the \textit{weak amenability constant} $\mathrm{WAM}(A)$, which is an analogue of the amenability constant $\mathrm{AM}(A)$ defined in [Runde, 2020, Definition 2.2.7]. I will present certain properties of $\mathrm{WAM}(A)$ and compute its values for certain noncommutative Banach algebras of complex $2 \times 2$ matrices equipped with different norms.

\noindent\textbf{References}

K. Koczorowski and K. Piszczek (2024), \textit{Weak Amenability of Banach Algebra-Valued $\ell_p$-Sequence Algebras.} Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 118, 89.

V. Runde (2020), \textit{Amenable Banach Algebras. A Panorama.} Springer Monographs in Mathematics. Springer Science + Business Media, LLC.