Factorization of homomorphisms through \(H^{\infty}\)(\(D\))
type: Item
instance of: scholarly article
title: Factorization of homomorphisms through \(H^{\infty}\)(\(D\))
zbMATH Open document ID: 1029.46070
DOI: 10.1016/S0022-247X(03)00065-9
author: Pablo Galindo
author: Mikael Lindström
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-05-27T00:00:00Z
review text: 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.
reviewed by: Hermann Render
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 47B33
zbMATH DE Number: 1917780
zbMATH Keywords: compact homomorphism
zbMATH Keywords: composition operator
zbMATH Keywords: URM-algebra
zbMATH Keywords: Gleason part
zbMATH Keywords: factorization
MaRDI profile type: Publication
cites work: Q4347432
cites work: An example of a nuclear space in infinite dimensional holomorphy
cites work: $H^{∞}$ is a Grothendieck space
cites work: Q4257163
cites work: Q4700286
cites work: Gleason parts and weakly compact homomorphisms between uniform Banach algebras
cites work: WEAKLY COMPACT HOMOMORPHISMS BETWEEN SMALL ALGEBRAS OF ANALYTIC FUNCTIONS
cites work: Q5615632
cites work: Q3922907
cites work: Bounded analytic functions and Gleason parts
cites work: Q3345273
cites work: Compact operators of the form \(uC_\varphi\)
cites work: Analytic maps of the open unit disk onto a Gleason part
cites work: Q4375183
cites work: Zur abstrakten Theorie der analytischen Funktionen. II.
cites work: Analytic functions and Dirichlet problem
cites work: Components in the space of composition operators
cites work: Topological structure of the space of composition operators on \(H^\infty\).
cites work: On composition operators on Banach spaces of holomorphic functions
cites work: Q3711155
cites work: Linearization of Bounded Holomorphic Mappings on Banach Spaces
cites work: Isolation amongst the composition operators
full work available at URL: s0022-247x(03)00065-9
OpenAlex ID: W1974399123
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description: scientific article; zbMATH DE number 1917780