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

Permanent link to this document
https://projecteuclid.org/euclid.nihmj/1571277623

Mathematical Reviews number (MathSciNet)
MR4019889

Zentralblatt MATH identifier
07155344

Subjects
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

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

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


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References

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