## Nihonkai Mathematical Journal

### Classification of semisimple commutative Banach algebras of type I

#### Abstract

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.

#### Note

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.

#### Note

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

#### Article information

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

Dates
Revised: 5 April 2019
First available in Project Euclid: 17 October 2019

https://projecteuclid.org/euclid.nihmj/1571277623

Mathematical Reviews number (MathSciNet)
MR4019889

Zentralblatt MATH identifier
07155344

#### Citation

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. https://projecteuclid.org/euclid.nihmj/1571277623

#### References

• 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.