On interpolation of Banach algebras and factorization of weakly compact homomorphisms
The family \((\cdot,\cdot)_\Gamma\) of interpolation methods was introduced by \textit{J.\,Peetre} [``A theory of interpolation of normed spaces'' (Notas Mat.\ No.\ 39) (1968; Zbl 0162.44502)]. A special class of the family \(\Gamma\) was used in [\textit{A.\,Blanco, S.\,Kaijser} and \textit{T.\,J.\thinspace Ransford}, J.~Funct.\ Anal.\ 217, No.\,1, 126--141 (2004; Zbl 1078.46050)] to prove that weakly compact homomorphisms between Banach algebras factor through a reflexive Banach algebra. It is crucial in this factorization that the interpolation method preserves the Banach algebra structure. In the paper under review, a necessary and sufficient condition on \(\Gamma\) for the interpolation method \((\cdot,\cdot)_\Gamma\) to preserve the Banach algebra structure is found. Applying this general result, it turns out that the classical real interpolation method \((\cdot,\cdot)_{\theta,q}\) preserves Banach algebras only if \(q=1\). Also, some results on the factorization of weakly compact homomorphisms between Banach algebras are derived as applications.
scientific article; zbMATH DE number 5128466