This entry is to try understanding the article
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
which asserts that for a monoidal subcategory of the category of unitary representations of a locally compact group to be sufficient for the reconstruction from the fiber functor , it suffices that contains the regular representation of on where is the left Haar measure on .
Table of contents
Here monoidal categories, monoidal functors and their morphisms are reviewed and (antilinear) -involutions on -linear monoidal categories. (Small) monoidal -categories form a category . For any (monoidal) -category, and any group , the category of representation of in objects in has a structure of a (monoidal) -category and the forgetful functor is a strict monoidal functor of -categories.
The case of abstract groups
Here a basic pair of adjoint functors is defined where gives the group of automorphisms of a monoidal -functor with codomain and its left adjoint is .
The case of topological groups
Here one supposes that is enriched over the category of topological (complex) vector spaces, so that is a topological group for all . The construction of an adjoint pair of functors above works with the category replaced by . However, in the representation theory one assumes only that the representations are continuous in the weak operator topology; and the group of invertibles in 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.
Reflexive monoidal categories are defined: they have an involution and functorial isomorphisms for all objects . 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.
The reconstruction problem
From now on a full subcategory 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 which are injective and with dense image in weak topology on the tensor product. Given a topological group , one chooses a full (monodial) -subcategory of the -category of continuous uniformly bounded representations of in . The group of automorphisms of the forgetful functor is equipped with weak operator topology and the canonical morphism is continuous; however is not a topological group in general. The reconstruction problem is, under which conditions on , the canonical isomorphism is an isomorphism of topological groups.
The algebra of matrix elements
One considers the set of matrix elements of representations belonging to the subcategory . It appears to be a subalgebra of the Banach -algebra of complex valued continuous functions on , and is closed with respect to involution , , closed under complex conjugation (which is an anti-involution) and an -submodule. It is proved that each elements of determine an automorphism of the algebra with involution .
subset of the algebra of bounded
Conditions of continuity
Here certain lemmas on certain category of algebras with involutions and anti-involutions is proved.
Among several theorems, the theorem 5.3 states that if is locally compact then a subcategory then for a -category of uniformly bounded continuous representations the canonical map to be an isomorphism of topological groups it is sufficient that the following simultaneously hold:
it separates the elements of
the algebra of matrix elements has nonzero intersection with for some , where is a left-invariant measure on
Classical corollaries are also shown. For example, for Tannaka-Krein one notes that all matrix elements of a compact group are in , 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 and uses stronger theorem 5.1 from the paper. Pontrjagin duality is a known corollary of Tatsuuma’s duality theorem.