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