BSE property of Fréchet algebra. (English) Zbl 1540.46042
Summary: A class of commutative Banach algebras which satisfy a Bochner-Schoenberg-Eberlein-type inequality was introduced by
S.-E. Takahasi and O. Hatori [Proc. Am. Math. Soc. 110, No. 1, 149–158 (1990; Zbl 0722.46025)]. We generalize this property for the commutative Fréchet algebra \((\mathscr{A}, p_\ell)_{\ell \in \mathbb{N}}\). Moreover, we verify and generalize some of the main results in the class of Banach algebras, for the Fréchet case. We prove that all Fréchet \(\mathrm{C}^*\)-algebras and also uniform Fréchet algebras are BSE algebras. Also, we show that \(C^\infty [0,1]\) is not a Fréchet BSE algebra.
MSC:
| 46J05 | General theory of commutative topological algebras |
| 46J10 | Banach algebras of continuous functions, function algebras |
| 46A04 | Locally convex Fréchet spaces and (DF)-spaces |
| 46M40 | Inductive and projective limits in functional analysis |
| 47L40 | Limit algebras, subalgebras of \(C^*\)-algebras |
Keywords:
BSE algebra; BSE function; commutative Fréchet algebra; Fréchet \(\mathrm{C}^*\)-algebra; multiplier algebra; uniform Fréchet algebraCitations:
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