Reconstruction of Groups

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aimed at a reasonably general reconstruction theorem for locally compact topological groups from a suitable category of continuous infinite-dimensional representations, generalizing the Pontrjagin duality theorem for locally compact abelian groups and the Tannaka-Krein theorem for compact groups. More precisely, it asks

what are conditions on a monoidal subcategory of the category of continuous representations of a locally compact group sufficient to reconstruct the group?

The first generalization of that type is the Tatsuuma reconstruction theorem

  • N. Tatsuuma, A duality theorem for locally compact groups, J. Math., Kyoto Univ. 6 (1967), 187–293.

which asserts that for a monoidal subcategory CC of the category of unitary representations of a locally compact group GG to be sufficient for the reconstruction from the fiber functor CHilbC\to Hilb, it suffices that CC contains the regular representation of GG on L 2(G,μ)L^2(G,\mu) where μ\mu is the left Haar measure on GG.

Table of contents

1.1. *\ast-categories, *\ast-functors

Here monoidal categories, monoidal functors and their morphisms are reviewed and (antilinear) *\ast-involutions on \mathbb{C}-linear monoidal categories. (Small) monoidal *\ast-categories form a category Cat *Cat^\ast. For any (monoidal) *\ast-category, and any group GG, the category Rep G(C)Rep^G(C) of representation of GG in objects in CC has a structure of a (monoidal) *\ast-category Rep G(C) *˜Rep^G(C)^{\tilde\ast} and the forgetful functor Rep G(C)CRep^G(C)\to C is a strict monoidal functor of *\ast-categories.

2.2. Duality construction

2.12.1 The case of abstract groups

Here a basic pair of adjoint functors is defined C˜ *Aut(C˜ *)\mathcal{R}_{C\tilde{}^\ast}\dashv Aut(C\tilde{}^\ast) where Aut(C˜ *):Cat */C˜ *GroupAut(C\tilde{}^\ast): Cat^\ast/C\tilde{}^\ast\to Group gives the group of automorphisms of a monoidal *\ast-functor with codomain C˜ *C\tilde{}^\ast and its left adjoint is C˜ *:GRep G(C)˜ *\mathcal{R}_{C\tilde{}^\ast} : G\mapsto Rep^G(C)\tilde{}^\ast.

2.22.2 The case of topological groups

Here one supposes that CC is enriched over the category of topological (complex) vector spaces, so that Aut(V)Aut(V) is a topological group for all VObCV\in Ob C. The construction of an adjoint pair of functors above works with the category GroupGroup replaced by TopGroupTopGroup. However, in the representation theory one assumes only that the representations are continuous in the weak operator topology; and the group of invertibles in EndVEnd V in the enriched sense is not always a topological group in that topology. Instead one works with the category of groups with a topology in which only left and right translations are required to be continuous.

2.32.3 Reflexivity

Reflexive monoidal categories are defined: they have an involution σ\sigma and functorial isomorphisms σ(X)XId\sigma(X)\otimes X\to Id for all objects XX. For example, rigid symmetric monodial categories and rigid braided monoidal categories are in that class; the category of Hilbert spaces and if a category is reflexive than the category of representations of a group is reflexive.

2.42.4 The reconstruction problem

From now on a full subcategory C˜C\tilde{} of the category of reflexive locally convex topological complex vector spaces is fixed with a real structure; it is equipped with a monoidal product such that the forgetful functor to vector spaces is monoidal with help of maps F(V) F(W)F(VW)F(V)\otimes_{\mathbb{C}} F(W)\to F(V\otimes W) which are injective and with dense image in weak topology on the tensor product. Given a topological group GG, one chooses a full (monodial) *\ast-subcategory ˜\mathfrak{C}\tilde{} of the *\ast-category of continuous uniformly bounded representations of GG in C˜C\tilde{}. The group of automorphisms of the forgetful functor F :˜VecF_{\mathfrak{C}} : \mathfrak{C}{\tilde{}}\to Vec is equipped with weak operator topology and the canonical morphism GAut *(F )G\to Aut^\ast(F_{\mathfrak{C}}) is continuous; however Aut *(F )Aut^\ast(F_{\mathfrak{C}}) is not a topological group in general. The reconstruction problem is, under which conditions on ˜\mathfrak{C}{\tilde{}}, the canonical isomorphism is an isomorphism of topological groups.

3.3. The algebra of matrix elements

One considers the set A(˜)A(\mathfrak{C}\tilde{}) of matrix elements of representations belonging to the subcategory A(˜)A(\mathfrak{C}\tilde{}). It appears to be a subalgebra of the Banach *\ast-algebra of complex valued continuous functions on GG, and is closed with respect to involution σ:ff() 1\sigma: f\to f\circ (-)^{-1}, f σ(x)=f(x 1)f^\sigma(x) = f(x^{-1}), closed under complex conjugation (which is an anti-involution) and an M(G)M(G)-submodule. It is proved that each elements of Aut *(F ˜)Aut^\ast(F_{\mathfrak{C}\tilde{}}) determine an automorphism of the algebra with involution A()A(\mathfrak{C}).

subset of the algebra of bounded

4.4. Conditions of continuity

Here certain lemmas on certain category B𝔄B\mathfrak{A} of algebras with involutions and anti-involutions is proved.

5.5. Reconstruction theorems

Among several theorems, the theorem 5.3 states that if GG is locally compact then a subcategory ˜\mathfrak{C}\tilde{} then for a *\ast-category of uniformly bounded continuous representations the canonical map GAut *(F ˜)G\to Aut^\ast(F_{\mathfrak{C}\tilde{}}) to be an isomorphism of topological groups it is sufficient that the following simultaneously hold:

  • it separates the elements of GG

  • the algebra A()A(\mathfrak{C}) of matrix elements has nonzero intersection with L p(G,μ)L^p(G,\mu) for some p>0p\gt 0, where μ\mu is a left-invariant measure on GG

Classical corollaries are also shown. For example, for Tannaka-Krein one notes that all matrix elements of a compact group are in L 1(G,μ)L^1(G,\mu), hence only the separation property is needed. For Tatsuuma duality one notes that the matrix elements of the regular representation are convolutions of functions from L 2(G,μ)L^2(G,\mu) and uses stronger theorem 5.1 from the paper. Pontrjagin duality is a known corollary of Tatsuuma’s duality theorem.

Revised on July 12, 2011 14:28:05 by David Corfield (