[Paper Review] The Dixmier-Malliavin Theorem and Bornological Vector Spaces

Gal Dor|arXiv (Cornell University)|Jan 16, 2020

Homotopy and Cohomology in Algebraic Topology6 references4 citations

TL;DR

This paper reformulates the Dixmier-Malliavin theorem in the language of bornological vector spaces, demonstrating that the Gårding space of a smooth $G$-representation is not only an essential $C_c^inity(G)$-module but also a smooth one—meaning the canonical map $C_c^inity(G) \otimes_{C_c^inity(G)} M \to M$ is an isomorphism without completion. This strengthens Meyer's equivalence by eliminating the need for completed tensor products, yielding a cleaner, more categorical formulation.

ABSTRACT

This note is intended to reformulate the Dixmier-Malliavin theorem about smooth group representations in the language of bornological vector spaces, instead of topological vector spaces. This language turns out to allow a more general theorem to be proven, and we are able to use it to strengthen a result of Meyer from arXiv:math/0310094. This paper is based on a part of the author's thesis.

Motivation & Objective

• To reframe the Dixmier-Malliavin theorem in the language of bornological vector spaces instead of topological vector spaces.
• To eliminate technical redundancies arising from topological formulations, particularly the use of completed tensor products.
• To show that essential $C_c^\infty(G)$-modules are automatically smooth, i.e., the uncompleted tensor product suffices.
• To establish a cleaner categorical equivalence between smooth $G$-modules and smooth $C_c^\infty(G)$-modules.
• To advocate for bornological vector spaces as a more natural framework in representation theory, especially for Lie groups.

Proposed method

• Use the framework of bornological vector spaces to replace topological vector spaces in the formulation of smooth group representations.
• Define quasi-unital rings in the category $\mathcal{B}\mathrm{orn}$ of bornological vector spaces, generalizing the notion from topological algebras.
• Introduce the concept of smooth $C_c^\infty(G)$-modules via the uncompleted tensor product $C_c^\infty(G) \otimes_{C_c^\infty(G)} M \to M$ being an isomorphism.
• Prove that $C_c^\infty(G)$ is quasi-unital in $\mathcal{B}\mathrm{orn}$, ensuring the tensor product condition holds without completion.
• Use the Gårding functor to construct a $C_c^\infty(G)$-module that is both essential and smooth, via local sections and diagram chasing in the bornological setting.
• Perform a diagram chase using locally defined sections $s_0$, $s_1$ to construct a section $\eta$ into $R \otimes R' \otimes R'$, proving exactness in the complex $R' \otimes R' \otimes R' \rightrightarrows R' \otimes R' \to R'$.

Experimental results

Research questions

• RQ1 Can the Dixmier-Malliavin theorem be reformulated in a way that avoids the use of completed tensor products in the definition of essential modules?
• RQ2 Is the canonical map $C_c^\infty(G) \otimes_{C_c^\infty(G)} M \to M$ an isomorphism for all essential $C_c^\infty(G)$-modules $M$ in the bornological setting?
• RQ3 Does the notion of smoothness in the bornological category coincide with essentiality, thereby unifying the two concepts?
• RQ4 Can the equivalence between smooth $G$-modules and essential $C_c^\infty(G)$-modules be strengthened to an equivalence with smooth $C_c^\infty(G)$-modules?
• RQ5 Is the ring $C_c^\infty(G)$ quasi-unital in the category of bornological vector spaces, ensuring the tensor product condition holds without completion?

Key findings

• The Gårding space of a smooth $G$-representation is not only an essential $C_c^\infty(G)$-module but also a smooth $C_c^\infty(G)$-module, meaning the uncompleted tensor product map $C_c^\infty(G) \otimes_{C_c^\infty(G)} M \to M$ is an isomorphism.
• The completed tensor product is unnecessary in Meyer's equivalence: essential $C_c^\infty(G)$-modules are automatically smooth, so $\widehat{\otimes}$ can be replaced with $\otimes$.
• The ring $C_c^\infty(G)$ is quasi-unital in the category of bornological vector spaces, satisfying $C_c^\infty(G) \otimes_{C_c^\infty(G)} C_c^\infty(G) \xrightarrow{\sim} C_c^\infty(G)$.
• The proof relies on constructing a local section $\eta$ into $R \otimes R' \otimes R'$ using higher-order sections $s_0$ and $s_1$, ensuring exactness in the complex $R' \otimes R' \otimes R' \rightrightarrows R' \otimes R' \to R'$.
• The reformulation yields a cleaner categorical equivalence: smooth $G$-modules are equivalent to smooth $C_c^\infty(G)$-modules, without technical overhead from completion.
• The bornological framework reveals a deeper structural simplicity in the Dixmier-Malliavin theorem, suggesting it is inherently bornological in nature rather than topological.