Isomorphism Problem in a Special Class of Banach Function Algebras and its Application

Year 2021, , 305 - 317, 16.09.2021

Sın-eı Takahası Kiyoshi Shirayanagi , Makoto Tsukada

https://doi.org/10.33205/cma.952056

Cited By: 2

Abstract

Given a weight function τ, we introduce a new class of Banach function algebras with respect to τ,
denoted by C_0b(X, τ ). We provide a complete solution to the isomorphism problem in this class. We further characterize the BSE-extension and the Inoue-Doss ideal associated with it. As an application of our results, we show the
equivalence of the four statements: (i) C_0b(X, τ) is of BSE, (ii) C_0b(X, τ) is of BED, (iii) C_0b(X, τ) is Tauberian and (iv)
τ is bounded.

Keywords

Banach function algebra of type I, BSE-extension, Inoue-Doss ideal, BSE-algebra, BED-algebra, Tauberian Banach algebra

Thanks

This work was supported by the Research Institute for Mathematical Sciences, a Joint Usage/Research Center located in Kyoto University.

References

• R. Doss: On the transform of a singular or an absolutely continuous measure, Proc. Amer. Math. Soc., 19 (1968), 361-363.
• J. Inoue, T. Miura, H. Takagi and S.-E. Takahasi: Classification of semisimple commutative Banach algebras of type I, Nihonkai Math. J., 30 (1) (2019), 1-17.
• J. Inoue, S.-E. Takahasi: Constructions of bounded weak approximate identities for Segal algebras on LCA groups, Acta Sci. Math., 66 (2000), 257-271.
• J. Inoue, S.-E. Takahasi: On characterizations of the image of the Gelfand transform of commutative Banach algebras, Math. Nachr., 280 (2007), 105-126.
• J. Inoue, S.-E. Takahasi: Segal algebras in commutative Banach algebras, Rocky Mountain J. Math., 44 (2) (2014), 539-589.
• J. Inoue, S.-E. Takahasi: A construction of a BSE-algebra of type I which is isomorphic to no C∗-algebras, Rocky Mountain J. Math., 47 (8) (2017), 2693-2697.
• J. Inoue, S.-E. Takahasi: Banach function algebras of n-times continuously differentiable functions on Rd vanishing at infinity and their BSE-extensions, J. Korean Math. Soc., 56 (5) (2019), 1333-1354.
• C. A. Jones, C. D. Lahr: Weak and norm approximate identities are different, Pac. J. Math., 72 (1977), 99-104.
• E. Kaniuth, A. Ülger: The Bochner-Schoenberg-Eberlein property for commutative Banach algebras, especially Fourier and Fourier Stieltjes algebras, Trans. Amer. Math. Soc., 362 (2010), 4331-4356.
• H. Reiter: L1-algebras and Segal algebras, Lect. Notes Math., 231, Springer-Verlag, Berlin (1971).
• H. Reiter, J. D. Stegeman: Classical Harmonic Analysis and Locally compact groups, Oxford Science Publications, Oxford (2000).
• C. E. Rickart: General Theory of Banach Algebras, D. Van Nostrand Company, Inc. Princeton, New Jersey, Toronto, London, New York (1960).
• S.-E. Takahasi, O. Hatori: Commutative Banach algebras which satisfy a Bochner-Schoenberg-Eberlein type-theorem, Proc. Amer. Math. Soc., 110 (1) (1990), 149-158.