2024/01/11 by Oleg Aristov, Aristov, O. Yu.
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)
paper · pdf · doi:10.48550/arxiv.2401.10199
openalex publication_date 2024/01/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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^∞-functional calculus for every element, we get a functor that maps the algebra of polynomials in k variables to the algebra of C^∞-functions on \Rk. The envelope of a general commutative or non-commutative algebra can be treated as an algebra of C^∞-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^∞-functions', i.e., the envelopes of free associative algebras of finite rank k, is announced for general k and proved for k≤ 2.