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Reconstruction of Groups

This entry is to try understanding the article

• A. L. Rosenberg, Reconstruction of groups, Selecta Math. N.S. 9:1 (2003) doi

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 $C$ of the category of unitary representations of a locally compact group $G$ to be sufficient for the reconstruction from the fiber functor $C\to \mathrm{Hilb}$, it suffices that $C$ contains the regular representation of $G$ on $L^2(G,\mu )$ where $\mu$ is the left Haar measure on $G$.

Table of contents

$1.$ $*$-categories, $*$-functors

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

$2.$ Duality construction

$2.1$ The case of abstract groups

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

$2.2$ The case of topological groups

Here one supposes that $C$ is enriched over the category of topological (complex) vector spaces, so that $\mathrm{Aut}(V)$ is a topological group for all $V\in \mathrm{Ob}C$. The construction of an adjoint pair of functors above works with the category $\mathrm{Group}$ replaced by $\mathrm{TopGroup}$. However, in the representation theory one assumes only that the representations are continuous in the weak operator topology; and the group of invertibles in $\mathrm{End}V$ 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.

$2.3$ Reflexivity

Reflexive monoidal categories are defined: they have an involution $\sigma$ and functorial isomorphisms $\sigma (X)\otimes X\to \mathrm{Id}$ for all objects $X$. 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.

$2.4$ The reconstruction problem

From now on a full subcategory $C\tilde{}$ 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 $F(V){\otimes }_{ℂ}F(W)\to F(V\otimes W)$ which are injective and with dense image in weak topology on the tensor product. Given a topological group $G$, one chooses a full (monodial) $*$-subcategory $ℭ\tilde{}$ of the $*$-category of continuous uniformly bounded representations of $G$ in $C\tilde{}$. The group of automorphisms of the forgetful functor $F_{ℭ}:ℭ\tilde{}\to \mathrm{Vec}$ is equipped with weak operator topology and the canonical morphism $G\to {\mathrm{Aut}}^{*}(F_{ℭ})$ is continuous; however ${\mathrm{Aut}}^{*}(F_{ℭ})$ is not a topological group in general. The reconstruction problem is, under which conditions on $ℭ\tilde{}$, the canonical isomorphism is an isomorphism of topological groups.

$3.$ The algebra of matrix elements

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

subset of the algebra of bounded

$4.$ Conditions of continuity

Here certain lemmas on certain category $B𝔄$ of algebras with involutions and anti-involutions is proved.

$5.$ Reconstruction theorems

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

• it separates the elements of $G$

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

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