## Nihonkai Mathematical Journal

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

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

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

#### 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