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The Dixmier-Malliavin Theorem and Bornological Vector Spaces

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The Dixmier-Malliavin Theorem and Bornological Vector Spaces The Dixmier-Malliavin Theorem and Bornological Vector Spaces Gal Dor Tel-Aviv University January 2020 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 [Mey03b]. This paper is based on a part of the author's thesis [Dor20]. Contents 1 Introduction 1 2 Bornological Vector Spaces 4 2.1 Background . .4 2.2 Tensor Products and Modules . .7 3 Quasi-Unital Rings 8 3.1 Definition of Quasi-Unital Rings in Vect . .8 3.2 Definition of Quasi-Unital Rings in Born . 10 3.3 Properties of Quasi-Unital Rings . 12 4 The G˚ardingFunctor 13 arXiv:2001.05694v1 [math.RT] 16 Jan 2020 5 Proof of Dixmier-Malliavin 15 A Proof of Claim 3.20 17 1 Introduction In this text, we will discuss bornological vector spaces and prove a variant of a theorem by Dixmier-Malliavin (see [Cas11] for a clean overview of the origi- 1 nal). Along the way, we will discuss the notion of quasi-unital algebra, which frequently comes up in the representation theory of locally compact groups. The first of these topics, bornological vector spaces, will be used in much of this text as a substitute for topological vector spaces. The two notions are very closely related (to the point that in many applications, they are indistin- guishable), but sometimes that leads to theorems that should be stated in one language being stated in the other. As a result, some theorems gain spurious technical requirements that disappear once they are stated correctly. It seems that many theorems in representation theory are much cleaner when stated in the language of bornological vector spaces instead of topological vector spaces. As a result, this text was written with a hidden agenda in mind: help push more people working with representation theory into using this language. Remark 1.1. It is fairly common to need to place some subcategory of topo- logical vector spaces into a \nice" categorical framework (usually, this means a locally presentable category). Every application has a different suitable choice of framework. For example, Scholze works with condensed sets in [Sch], a con- struction inspired by the pro-´etalesite. It seems to the author that bornological vector spaces are useful in the con- text of representation theory beyond simply being a replacement for the category of topological vector spaces with nicer formal properties. This case is heavily advocated by Meyer, e.g. in [Mey03b, Mey04, Mey05, Mey03a]. However, most of his motivation comes from the world of non-commutative geometry and ho- mological algebra. The current text is focused on more classical motivations, by showing that the Dixmier-Malliavin theorem has a distinctly bornological “flavor". This brings us to the second topic we discuss in this paper: the Dixmier- Malliavin theorem. The content of our version of this theorem is that the 8 G˚ardingspace of a complete bornological representation is a smooth Cc pGq- module. Classically, as it is formulated in [Cas11], the theorem is restricted to Fr´echet topological representations, which are a full subcategory of the bornolog- ical representations. Much of this text does not constitute original contribution. Certainly, ev- erything we have to say about bornological vector spaces or spaces of smooth functions is already known, and is included here only as a reference. Many of our claims about quasi-unital algebras already appear in [Mey09]. Our contri- bution to Dixmier-Malliavin is its reformulation in terms of bornological vector spaces, which results in a cleaner and more categorical “flavor". 8 Let us give some more details. Let G be a real Lie group. Recall that Cc pGq is the (bornological) space of smooth and compactly supported functions on G, and that convolution turns it into a ring. In [Mey03b], Meyer proves an equivalence between two categories: 8 tsmooth G-modulesu o / tessential Cc pGq-modulesu: On the left hand side of the equivalence, is the category of complete bornolog- ical G-modules, where the action of G is smooth (i.e., its G˚ardingspace is equal 2 to itself, see also Definition 4.6). On the right hand side of Meyer's equivalence 8 is the full subcategory of complete bornological Cc pGq-modules M satisfying that the canonical morphism: 8 C G 8 M M c p qbCc pGq Ñ is an isomorphism. Here, b is the completionp of the relative (projective) tensor 8 product. Meyer refers to this category as the category of essential Cc pGq- modules. p Note that Meyer's notion of essential module makes sense because Meyer 8 shows that the ring Cc pGq satisfies this property over itself, i.e.: 8 8 „ 8 C G 8 C G C G : c p qbCc pGq c p q ÝÑ c p q We will prove a strengtheningp of this statement (although, for simplicity, we consider less general groups G compared to [Mey03b]). Specifically, we will show that the use of the completed tensor product is unnecessary. More explicitly, 8 all essential Cc pGq-modules M also satisfy that the morphism 8 C G 8 M M c p q bCc pGq Ñ is already an isomorphism of bornological vector spaces { that is, no completion 8 is necessary. We will refer to this property as smoothness of a Cc pGq-module. 8 Once again, this property will make sense because we will show that Cc pGq is in fact quasi-unital, i.e.: 8 8 „ 8 C G 8 C G C G : c p q bCc pGq c p q ÝÑ c p q For more on our definitions of quasi-unitality and smoothness, we refer the reader to Section 3. Note that our terminology differs somewhat from that of [Mey09]. 8 In short, we claim that essential Cc pGq-modules are automatically smooth 8 Cc pGq-modules, and establish an equivalence: 8 tsmooth G-modulesu o / tsmooth Cc pGq-modulesu: (1) In order to prove this remarkable property, we will use a strengthened version of the theorem of Dixmier-Malliavin. Specifically, we will show that the G˚arding 8 functor defined in Section 4 defines an essential Cc pGq-module which is also a 8 smooth Cc pGq-module. This will also allow us to prove some useful corollaries. Remark 1.2. Our notion of smoothness is such that a smooth bornological 8 Cc pGq-module always satisfies that 8 C G 8 M M c p q bCc pGq Ñ is an isomorphism of vector spaces (without any topology or bornology). Thus, our version of the Dixmier-Malliavin theorem implies the classical one. 3 The structure of this text is as follows. In Section 2, we will recall the notion of a bornological vector space and its basic properties. Additionally, we will linger on the comparison between bornological and topological vector spaces. In Section 3, we will recall the notions of a quasi-unital algebra and its smooth modules. In Section 4, we will define the G˚arding functor on bornological rep- resentations, which will allow us to define the notion of a smooth G-module. Armed with a definition for the two categories involved in the equivalence (1), in Section 5 we will finally prove the desired equivalence, and establish the Dixmier- Malliavin theorem in the bornological setting. In Appendix A, we will prove a result from Section 3 that is not needed for the proof of our main theorem. Acknowledgements: The author would like to thank his advisor, Joseph Bernstein, for pointing out the Dixmier-Malliavin theorem to him, and his great help in refining this paper. The author would also like to thank Shachar Carmeli for many fruitful discussions about the category of bornological vector spaces. 2 Bornological Vector Spaces In this section, we will introduce some background knowledge on bornological vector spaces for the reader who is not necessarily familiar with their use, and try to motivate their relevance to our work. This section should not be considered original contribution. This section is far too short to be a comprehensive introduction to the theory of bornological vector spaces. However, the author hopes to emphasize both the added convenience that they give over using topological vector spaces, and their intuitive sense. We will begin with a general background on bornological vector spaces and their comparison with topological vector spaces. Afterwards, we will discuss bornological modules, and show some of their advantages over topological mod- ules. Warning 2.1. The reader should beware that in this text, we are not auto- matically assuming that our bornological vector spaces are complete. Likewise, we will take care to distinguish between the (incomplete) tensor product b, and the completed tensor product b. 2.1 Background p Generally speaking, just like a topological space specifies which of its subsets is open, a bornological space is supposed to specify which of its subsets is bounded. While in most circumstances topological and bornological spaces are fairly dif- ferent notions, their specializations to the case of vector spaces are more closely related than might be expected. Let us begin by introducing some notation. Definition 2.2. We denote: 1. By Vect the category of vector spaces over C. 4 2. By SNorm the category of semi-normed topological vector spaces over C. 3. By Top the category of locally convex topological vector spaces over C. 4. By Born the category of bornological vector spaces, with convex vector bornologies over C.
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