Envelopes in the class of Banach algebras of polynomial growth and $C^\infty$-functions of a finite number of free variables
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866913948961341440 |
|---|---|
| author | Aristov, O. Yu. |
| author_facet | Aristov, O. Yu. |
| contents | We introduce the notion of envelope of a topological algebra (in particular, an arbitrary associative algebra) with respect to a class of Banach algebras. In the case of the class of real Banach algebras of polynomial growth, i.e., admitting a $C^\infty$-functional calculus for every element, we get a functor that maps the algebra of polynomials in $k$ variables to the algebra of $C^\infty$-functions on $\R^k$. The envelope of a general commutative or non-commutative algebra can be treated as an algebra of $C^\infty$-functions on some commutative or non-commutative space. In particular, we describe the envelopes of the universal enveloping algebra of finite-dimensional Lie algebras, the coordinate algebras of the quantum plane and quantum $SL(2)$ and also look at some commutative examples. A result on algebras of `free $C^\infty$-functions', i.e., the envelopes of free associative algebras of finite rank $k$, is announced for general $k$ and proved for $k\le 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_10199 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Envelopes in the class of Banach algebras of polynomial growth and $C^\infty$-functions of a finite number of free variables Aristov, O. Yu. Functional Analysis Operator Algebras We introduce the notion of envelope of a topological algebra (in particular, an arbitrary associative algebra) with respect to a class of Banach algebras. In the case of the class of real Banach algebras of polynomial growth, i.e., admitting a $C^\infty$-functional calculus for every element, we get a functor that maps the algebra of polynomials in $k$ variables to the algebra of $C^\infty$-functions on $\R^k$. The envelope of a general commutative or non-commutative algebra can be treated as an algebra of $C^\infty$-functions on some commutative or non-commutative space. In particular, we describe the envelopes of the universal enveloping algebra of finite-dimensional Lie algebras, the coordinate algebras of the quantum plane and quantum $SL(2)$ and also look at some commutative examples. A result on algebras of `free $C^\infty$-functions', i.e., the envelopes of free associative algebras of finite rank $k$, is announced for general $k$ and proved for $k\le 2$. |
| title | Envelopes in the class of Banach algebras of polynomial growth and $C^\infty$-functions of a finite number of free variables |
| topic | Functional Analysis Operator Algebras |
| url | https://arxiv.org/abs/2401.10199 |