Characterization of commutative algebras embedded into the algebra of smooth operators
| Autor: | Ciaś, Tomasz |
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| Rok vydání: | 2021 |
| Předmět: | |
| Zdroj: | Bull. Lond. Math. Soc. 49 (2017), no. 1, 102-116 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1112/blms.12010 |
| Popis: | The paper deal with the noncommutative Fr\'echet ${}^*$-algebra $\mathcal{L}(s',s)$ of the so-called smooth operators, i.e. linear and continuous operators acting from the space $s'$ of slowly increasing sequences to the Fr\'echet space $s$ of rapidly decreasing sequences. By a canonical identification, this algebra of smooth operators can be also seen as the algebra of the rapidly decreasing matrices. We give a full description of closed commutative ${}^*$-subalgebras of this algebra and we show that every closed subspace of $s$ with basis is isomorphic (as a Fr\'echet space) to some closed commutative ${}^*$-subalgebra of $\mathcal{L}(s',s)$. As a consequence, we give some equivalent formulation of the long-standing Quasi-equivalence Conjecture for closed subspaces of $s$. Comment: 14 pages |
| Databáze: | arXiv |
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