Factorization of homomorphisms through \(H^{\infty}\)(\(D\))
A uniform Banach algebra is called a (URM)-algebra if for every non-trivial Gleason part \(P\) each \( n\in P\) has a unique representing measure on the Shilov boundary. For example, every logmodular algebra is a (URM)-algebra, in particular the disk algebra and the algebra \(H^\infty (D) \) of all bounded analytic functions on the open unit disk \(D\) are in this class. As usual, \(M(A) \) denotes the maximal ideal space of \(A\) and \(\hat {f} \) the Gelfand transform of \( f.\) The main result of the paper is the following interesting description for weakly compact homomorphisms \(T\) from a (UMR)-algebra \(A\) into a uniform Banach algebra \(B\): there exist an element \( b \in B \) of norm \(<1\), mappings \( L_m : D\to M(A) \) and idempotents \( b_1,\dots,b_N \in B \) such that \( \widehat{Tf} =\sum_{m=1}^N \widehat{b} \cdot (\hat{f}\circ L_m\circ \hat{b}) . \) It follows that under the above assumptions a weakly compact homomorphism is strongly nuclear. In the case that \(M(B) \) is connected, the number \(N\) is equal to \(1 \) and the above result can be restated as a factorization of \(T\) by composition operators. It is also shown that the assumption of a (URM)-algebra is essential; in this context a characterization of strong nuclearity of composition operators on spaces of bounded analytic functions is given. Finally, it is shown that the path connected components of the space of all homomorphisms on \(H^\infty (D) \) are the open balls of radius 2.
scientific article; zbMATH DE number 1917780
scientific article