Multipliers with closed range on regular commutative Banach algebras

Aiena, Pietro; Laursen, Kjeld B.

doi:10.1090/S0002-9939-1994-1185257-1

Multipliers with closed range on regular commutative Banach algebras
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by Pietro Aiena and Kjeld B. Laursen

Proc. Amer. Math. Soc. 121 (1994), 1039-1048

DOI: https://doi.org/10.1090/S0002-9939-1994-1185257-1

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Abstract:

Conditions equivalent with closure of the range of a multiplier T, defined on a commutative semisimple Banach algebra A, are studied. A main result is that if A is regular then ${T^2}A$ is closed if and only if T is a product of an idempotent and an invertible. This has as a consequence that if A is also Tauberian then a multiplier with closed range is injective if and only if it is surjective. Several aspects of Fredholm theory and Kato theory are covered. Applications to group algebras are included.

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