Open Access
2019 Classification of semisimple commutative Banach algebras of type I
Jyunji Inoue, Takeshi Miura, Hiroyuki Takagi, Sin-Ei Takahasi
Nihonkai Math. J. 30(1): 1-17 (2019).


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.

Funding Statement

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


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.


Download Citation

Jyunji Inoue. Takeshi Miura. Hiroyuki Takagi. Sin-Ei Takahasi. "Classification of semisimple commutative Banach algebras of type I." Nihonkai Math. J. 30 (1) 1 - 17, 2019.


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

zbMATH: 07155344
MathSciNet: MR4019889

Primary: 46J10
Secondary: 46H20 , 46J40

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

Rights: Copyright © 2019 Niigata University, Department of Mathematics

Vol.30 • No. 1 • 2019
Back to Top