Let A and B be semisimple commutative Banach algebras with bounded approximate identities. We investigate the problem of extending a homomorphism φ: A → B to a homomorphism of the multiplier algebras M(A) and M(B) of A and B, respectively. Various sufficient conditions in terms of B (or B and φ) are given that allow the construction of such extensions. We exhibit a number of classes of Banach algebras to which these criteria apply. In addition, we prove a polar decomposition for homomorphisms from A into A with closed range. Our results are applied to Fourier algebras of locally compact groups.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm183-1-3,
author = {E. Kaniuth and A. T. Lau and A. \"Ulger},
title = {Homomorphisms of commutative Banach algebras and extensions to multiplier algebras with applications to Fourier algebras},
journal = {Studia Mathematica},
volume = {178},
year = {2007},
pages = {35-62},
zbl = {1152.46040},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm183-1-3}
}
E. Kaniuth; A. T. Lau; A. Ülger. Homomorphisms of commutative Banach algebras and extensions to multiplier algebras with applications to Fourier algebras. Studia Mathematica, Tome 178 (2007) pp. 35-62. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm183-1-3/