Envelopes in the class of Banach algebras of polynomial growth and $C^\infty$-functions of a finite number of free variables

TL;DR

This paper introduces the concept of envelopes of topological algebras with respect to Banach algebras of polynomial growth, linking algebraic structures to $C^ abla$-function spaces and exploring their applications in quantum and Lie algebra contexts.

Contribution

It develops a framework for envelopes of algebras as $C^ abla$-function spaces, including explicit descriptions for universal enveloping algebras and quantum groups, and proves results for free associative algebras.

Findings

Envelopes of universal enveloping algebras of Lie algebras are described.

Envelopes of quantum groups like quantum $SL(2)$ are characterized.

Results on free $C^ abla$-functions are established for low ranks.

Abstract

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 $k\le 2$.