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On interpolation of Banach algebras and factorization of weakly compact homomorphisms. (English) Zbl 1119.46024
The family \((\cdot,\cdot)_\Gamma\) of interpolation methods was introduced by 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 [A. Blanco, S. Kaijser and T. J. 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.

MSC:
46B70 Interpolation between normed linear spaces
46H05 General theory of topological algebras
46M35 Abstract interpolation of topological vector spaces
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