Multiplication operators on Banach modules over spectrally separable algebras

Document Type : Research Paper

Author

Department of Materials and Metallurgy‎, ‎Faculty of Natural Sciences and Engineering‎, ‎University of Ljubljana‎, ‎Aškerčeva c‎. ‎12‎, ‎SI-1000 Ljubljana‎, ‎Slovenia.

Abstract

‎Let $\mathcal{A}$ be a commutative Banach algebra and $\mathscr{X}$ be a left Banach $\mathcal{A}$-module‎. ‎We study the set‎ ‎${\rm Dec}_{\mathcal{A}}(\mathscr{X})$ of all elements in $\mathcal{A}$ which induce a decomposable multiplication operator on $\mathscr{X}$‎. ‎In the case $\mathscr{X}=\mathcal{A}$‎, ‎${\rm Dec}_{\mathcal{A}}(\mathcal{A})$ is the well-known Apostol algebra of $\mathcal{A}$‎. ‎We show that ${\rm Dec}_{\mathcal{A}}(\mathscr{X})$ is intimately related with the largest spectrally separable subalgebra of $\mathcal{A}$ and in this context‎ ‎we give some results which are related to an open question if Apostol algebra is regular for any commutative algebra $\mathcal{A}$‎.‎

Keywords

Main Subjects


E‎. ‎Albrecht‎, ‎Decomposable systems of operators in harmonic analysis‎, ‎in‎: ‎Toeplitz Centennial‎, ‎19--35‎, ‎(I‎. ‎Gohberg‎, ‎ed.)‎, ‎Birkhauser‎, ‎Basel‎, ‎1983‎.
C‎. ‎Apostol‎, Decomposable multiplication operators‎, Rev‎. ‎Roumaine Math‎. ‎Pures Appl. 17 (1972) 323--333‎.
A‎. ‎G‎. ‎Baskakov‎, ‎Spectral synthesis in Banach modules over commutative Banach algebras‎, Math‎. ‎Notes 34 (1983)‎, ‎no‎. ‎4‎, ‎776--782‎.
J‎. ‎Bračič‎, ‎Unital strongly harmonic commutative Banach algebras‎, Studia Math. 149 (2002)‎, ‎253--266‎.
J‎. ‎Bračič‎, ‎Strongly harmonic operators‎, Acta Sci‎. ‎Math. (Szeged) 68 (2002)‎, ‎797--813‎.
‎ I‎. ‎Colojoara and C‎. ‎Foias‎, ‎Theory of Generalized Spectral Operators‎,  ‎Gordon and Breach‎, ‎New York‎, ‎1968‎.
St‎. ‎Frunza‎, ‎A characterization of regular Banach algebras‎, Rev‎. ‎Roumaine Math‎. ‎Pures Appl. 18 (1973)‎, ‎1057--1059‎.
‎ J‎. ‎Inoue and S.-E‎. ‎Takahasi‎, ‎A note on the largest regular subalgebra of a Banach algebra‎, Proc‎. ‎Amer‎. ‎Math‎. ‎Soc. ‎ 116 (1992) 961--962‎.
‎ E‎. ‎Kaniuth‎, ‎A Course in Commutative Banach Algebras‎, ‎Graduate Texts in Mathematics 246‎, ‎Springer‎, ‎New York‎, ‎2009‎.
K‎. ‎B‎. ‎Laursen and M‎. ‎M‎. ‎Neumann‎, ‎An Introduction to Local Spectral Theory‎, London Math‎. ‎Soc‎. ‎Monographs 20‎, ‎Clarendon Press‎, ‎Oxford‎, ‎2000‎.
M‎. ‎M‎. ‎Neumann‎, ‎Commutative Banach algebras and decomposable operators‎, Monatsh‎. ‎Math. 113 (1992)‎, ‎227--243‎.
M‎. ‎M‎. ‎Neumann‎,‎Banach algebras‎, ‎decomposable convolution operators‎, ‎and a spectral mapping property‎, ‎in‎: ‎Function spaces‎, ‎(K‎. ‎Jarosz‎, ‎ed.)‎, ‎Marcel Dekker‎, ‎New York (1992)‎, ‎307--323‎.
C‎. ‎E‎. ‎Rickart‎, ‎General Theory of Banach Algebras‎,  ‎Van Nostrand‎, ‎Princeton 1960‎.