Nihonkai Mathematical Journal

Classification of semisimple commutative Banach algebras of type I

Jyunji Inoue, Takeshi Miura, Hiroyuki Takagi, and Sin-Ei Takahasi

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In the first and fourth authors' paper in 2017, it was shown that there exists a BSE-algebra of type I isomorphic to no C*-algebras, which solved negatively a question posed by the fourth author and O. Hatori. However, this result suggests a further investigation of commutative Banach algebra of type I. In the first part of the paper, we classify type I algebras into six families by means of BSE, BED, and Tauberian. It is shown that a Banach algebra of type I is isomorphic to a Segal algebra in some commutative C*-algebra if and only if it is Tauberian. In the second part, we give concrete examples of type I algebras to show that all of six families mentioned above are nonempty.


To our regret, the third author Prof. Hiroyuki Takagi passed away on November 25, 2017. We would like to express our deepest condolences for him.


Research of the second author was supported in part by JSPS KAKENHI Grant Number 15K04921 and 16K05172.

Article information

Nihonkai Math. J., Volume 30, Number 1 (2019), 1-17.

Received: 29 October 2018
Revised: 5 April 2019
First available in Project Euclid: 17 October 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46J10: Banach algebras of continuous functions, function algebras [See also 46E25]
Secondary: 46J40: Structure, classification of commutative topological algebras 46H20: Structure, classification of topological algebras

Banach algebra of type I BSE-algebra BED-algebra Segal algebra Tauberian algebra C*-algebra


Inoue, Jyunji; Miura, Takeshi; Takagi, Hiroyuki; Takahasi, Sin-Ei. Classification of semisimple commutative Banach algebras of type I. Nihonkai Math. J. 30 (2019), no. 1, 1--17.

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  • J. Inoue and S.-E. Takahasi, On characterizations of the image of the Gelfand transform of commutative Banach algebras, Math. Nachr., 280 (2007), 105–126.
  • J. Inoue and S.-E. Takahasi, Segal algebras in commutative Banach algebras, Rocky Mountain J. Math., 44-2 (2014), 539–589.
  • J. Inoue and 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.
  • E. Kaniuth and 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.
  • R. Larsen, An itroduction to the theory of multipliers, Springer-Verlag, New York, 1971.
  • H. Reiter, $L^1$-algebras and Segal algebras, Lect. Notes Math. 231, Springer-Verlag, Berlin, 1971.
  • H. Reiter and J.D. Stegeman, Classical Harmonic Analsis 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 and O. Hatori, Commutative Banach algebras which satisfy a Bochner-Schoenberg-Eberlein type-theorem, Proc. Amer. Math. Soc., 110-1 (1990), 149–158.