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The nicest way of phrasing it is the following. Let $\mathcal H$ be the category of Hilbert spaces with unitary maps between them. For each locally compact group $G$, one can define a functor $$Rep_G : {\mathcal H} \to Top$$ with $Rep_G(H) = \hom(G,U(H))$, where the space of homomorphisms is endowed with the compact-open topology (with respect to the strong operator topology on $U(H)$). Obviously, the functor $Rep_G$ is compatible with sums and tensor products of Hilbert spaces. Note that $Rep_{\mathbb Z}(H)=U(H).$

Consider now any such functor $F: {\mathcal H} \to Top$ and set $$D(F) = Nat_{\otimes,\oplus}(F,Rep_{\mathbb Z}),$$ i.e. all natural transformations of functors which are compatible with the tensor-product and the sum. $D(F)$ is a group since $Rep_{\mathbb Z}(H)=U(H)$ is a group for each Hilbert space $H$. It is also a topological group in a natural way.

Now, there is a natural map $\iota_G : G \to D(Rep_G)$ which is given by $\iota(g)(\pi) = \pi(g)$, where $\pi \in hom(G,U(H))$. So just as in the case of Pontrjagin duality, there is a natural bi-dual. A bit of work (relying on results of Takesaki and Gel'fand-Raikov (which you have mentioned)) shows that $\iota_G$ is a topological isomorphism for all locally compact topological groups.

I studied the analogous question which arises if one restricts everything to the category of finite-dimensional Hilbert spaces (see here). This sometimes goes under the name Chu duality, but is not so extensively studied. Everything works for locally compact abelian groups and compact groups by Pontrjagins result and the Tannaka-Krein theorem. However, for finitely generated discrete groups, interesting things happen. First of all, it is trivial to observe that the analogous map

$$\iota_G : G \to D_{fin}(Rep_G)$$

is injective if and only $G$ is maximally almost periodic (by a result of Mal'cev iff $G$ is residually finite). Moreover, and this is more difficult, $\iota_G$ is an isomorphism if and only if $G$ is virtually abelian. In particular, for $G={\mathbb F_2}$, the map $\iota_{\mathbb F_2}$ from $\mathbb F_2$ to $D_{fin}(Rep_{\mathbb F_2})$ is not surjective. This is a bit surprising as there are no natural candidates of elements in $D_{fin}(Rep_{\mathbb F_2})$, which do not lie in the image.

The nicest way of phrasing it is the following. Let $\mathcal H$ be the category of Hilbert spaces with unitary maps between them. For each locally compact group $G$, one can define a functor $$Rep_G : {\mathcal H} \to Top$$ with $Rep_G(H) = \hom(G,U(H))$, where the space of homomorphisms is endowed with the compact-open topology (with respect to the strong operator topology on $U(H)$). Obviously, the functor $Rep_G$ is compatible with sums and tensor products of Hilbert spaces. Note that $Rep_{\mathbb Z}(H)=U(H).$

Consider now any such functor $F: {\mathcal H} \to Top$ and set $$D(F) = Nat_{\otimes,\oplus}(F,Rep_{\mathbb Z}),$$ i.e. all natural transformations of functors which are compatible with the tensor-product and the sum. $D(F)$ is a group since $Rep_{\mathbb Z}(H)=U(H)$ is a group for each Hilbert space $H$. It is also a topological group in a natural way.

Now, there is a natural map $\iota_G : G \to D(Rep_G)$ which is given by $\iota_G(g)(\pi) = \pi(g)$, where $\pi \in hom(G,U(H))$. So just as in the case of Pontrjagin duality, there is a natural bi-dual. A bit of work (relying on results of Takesaki and Gel'fand-Raikov (which you have mentioned)) shows that $\iota_G$ is a topological isomorphism for all locally compact topological groups.

I studied the analogous question which arises if one restricts everything to the category of finite-dimensional Hilbert spaces (see here). This sometimes goes under the name Chu duality, but is not so extensively studied. Everything works for locally compact abelian groups and compact groups by Pontrjagins result and the Tannaka-Krein theorem. However, for finitely generated discrete groups, interesting things happen. First of all, it is trivial to observe that the analogous map

$$\iota_G : G \to D_{fin}(Rep_G)$$

is injective if and only $G$ is maximally almost periodic (by a result of Mal'cev iff $G$ is residually finite). Moreover, and this is more difficult, $\iota_G$ is an isomorphism if and only if $G$ is virtually abelian. In particular, for $G={\mathbb F_2}$, the map $\iota_{\mathbb F_2}$ from $\mathbb F_2$ to $D_{fin}(Rep_{\mathbb F_2})$ is not surjective. This is a bit surprising as there are no natural candidates of elements in $D_{fin}(Rep_{\mathbb F_2})$, which do not lie in the image.

The nicest way of phrasing it is the following. Let $\mathcal H$ be the category of Hilbert spaces with unitary maps between them. For each locally compact group $G$, one can define a functor $$Rep_G : {\mathcal H} \to Top$$ with $Rep_G(H) = \hom(G,U(H))$, where the space of homomorphisms is endowed with the compact-open topology (with respect to the strong operator topology on $U(H)$). Obviously, the functor $Rep_G$ is compatible with sums and tensor products of Hilbert spaces. Note that $Rep_{\mathbb Z}(H)=U(H).$

Consider now any such functor $F: {\mathcal H} \to Top$ and set $$D(F) = Nat_{\otimes,\oplus}(F,Rep_{\mathbb Z}),$$ i.e. all natural transformations of functors which are compatible with the tensor-product and the sum. $D(F)$ is a group since $Rep_{\mathbb Z}(H)=U(H)$ is a group for each Hilbert space $H$. It is also a topological group in a natural way.

Now, there is a natural map $\iota_G : G \to D(Rep_G)$ which is given by $\iota(g)(\pi) = \pi(g)$, where $\pi \in hom(G,U(H))$. So just as in the case of Pntrjagin duality, there is a natural bi-dual. A bit of work (relying on results of Takesaki and Gel'fand-Raikov (which you have mentioned)) shows that $\iota_G$ is a topological isomorphism for all locally compact topological groups.

I studied the analogous question which arises if one restricts everything to the category of finite-dimensional Hilbert spaces (see here). This sometimes goes under the name Chu duality, but is not so extensively studied. Everything works for locally compact abelian groups and compact groups by Pontrjagins result and the Tannaka-Krein theorem. However, for finitely generated discrete groups, interesting things happen. First of all, it is trivial to observe that the analogous map

$$\iota_G : G \to D_{fin}(Rep_G)$$

is injective if and only $G$ is maximally almost periodic (by a result of Mal'cev iff $G$ is residually finite). Moreover, and this is more difficult, $\iota_G$ is an isomorphism if and only if $G$ is virtually abelian. In particular, for $G={\mathbb F_2}$, the map $\iota_{\mathbb F_2}$ from $\mathbb F_2$ to $D_{fin}(Rep_{\mathbb F_2})$ is not surjective. This is a bit surprising as there are no natural candidates of elements in $D_{fin}(Rep_{\mathbb F_2})$, which do not lie in the image.

The nicest way of phrasing it is the following. Let $\mathcal H$ be the category of Hilbert spaces with unitary maps between them. For each locally compact group $G$, one can define a functor $$Rep_G : {\mathcal H} \to Top$$ with $Rep_G(H) = \hom(G,U(H))$, where the space of homomorphisms is endowed with the compact-open topology (with respect to the strong operator topology on $U(H)$). Obviously, the functor $Rep_G$ is compatible with sums and tensor products of Hilbert spaces. Note that $Rep_{\mathbb Z}(H)=U(H).$

Consider now any such functor $F: {\mathcal H} \to Top$ and set $$D(F) = Nat_{\otimes,\oplus}(F,Rep_{\mathbb Z}),$$ i.e. all natural transformations of functors which are compatible with the tensor-product and the sum. $D(F)$ is a group since $Rep_{\mathbb Z}(H)=U(H)$ is a group for each Hilbert space $H$. It is also a topological group in a natural way.

Now, there is a natural map $\iota_G : G \to D(Rep_G)$ which is given by $\iota(g)(\pi) = \pi(g)$, where $\pi \in hom(G,U(H))$. So just as in the case of Pontrjagin duality, there is a natural bi-dual. A bit of work (relying on results of Takesaki and Gel'fand-Raikov (which you have mentioned)) shows that $\iota_G$ is a topological isomorphism for all locally compact topological groups.

I studied the analogous question which arises if one restricts everything to the category of finite-dimensional Hilbert spaces (see here). This sometimes goes under the name Chu duality, but is not so extensively studied. Everything works for locally compact abelian groups and compact groups by Pontrjagins result and the Tannaka-Krein theorem. However, for finitely generated discrete groups, interesting things happen. First of all, it is trivial to observe that the analogous map

$$\iota_G : G \to D_{fin}(Rep_G)$$

is injective if and only $G$ is maximally almost periodic (by a result of Mal'cev iff $G$ is residually finite). Moreover, and this is more difficult, $\iota_G$ is an isomorphism if and only if $G$ is virtually abelian. In particular, for $G={\mathbb F_2}$, the map $\iota_{\mathbb F_2}$ from $\mathbb F_2$ to $D_{fin}(Rep_{\mathbb F_2})$ is not surjective. This is a bit surprising as there are no natural candidates of elements in $D_{fin}(Rep_{\mathbb F_2})$, which do not lie in the image.

The nicest way of phrasing it is the following. Let $\mathcal H$ be the category of Hilbert spaces with unitary maps between them. For each locally compact group $G$, one can define a functor $$Rep_G : {\mathcal H} \to Top$$ with $Rep_G(H) = \hom(G,U(H))$, where the space of homomorphisms is endowed with the compact-open topology (with respect to the strong operator topology on $U(H)$). Obviously, the functor $Rep_G$ is compatible with sums and tensor products of Hilbert spaces. Note that $Rep_{\mathbb Z}(H)=U(H).$

Consider now any such functor $F: {\mathcal H} \to Top$ and set $$D(F) = Nat_{\otimes,\oplus}(F,Rep_{\mathbb Z}),$$ i.e. all natural transformations of functors which are compatible with the tensor-product and the sum. $D(F)$ is a group since $Rep_{\mathbb Z}(H)=U(H)$ is a group for each Hilbert space $H$. It is also a topological group in a natural way.

Now, there is a natural map $\iota_G : G \to D(Rep_G)$ which is given by $\iota(g)(\pi) = \pi(g)$, where $\pi \in hom(G,U(H))$. So just as in the case of Pntrjagin duality, there is a natural bi-dual. A bit of work (relying on results of Takesaki and Gel'fand-Raikov (which you have mentioned)) shows that $\iota_G$ is a topological isomorphism for all locally compact topological groups.

I studied the analogous question which arises if one restricts everything to the category of finite-dimensional Hilbert spaces (see here). This some goes under the name Chu duality, but is not so extensively studied. Everything works for locally compact abelian groups and compact groups by Pontrjagins result and the Tannaka-Krein theorem. However, for finitely generated discrete groups, interesting things happen. First of all, it is trivial to observe that the analogous map

$$\iota_G : G \to D_{fin}(Rep_G)$$

is injective if and only $G$ is maximally almost periodic (by a result of Mal'cev iff $G$ is residually finite). Moreover, and this is more difficult, $\iota_G$ is an isomorphism if and only if $G$ is virtually abelian. In particular, for $G={\mathbb F_2}$, the map $\iota_{\mathbb F_2}$ from $\mathbb F_2$ to $D_{fin}(Rep_{\mathbb F_2})$ is not surjective. This is a bit surprising as they are no natural candidates of elements in $D_{fin}(Rep_{\mathbb F_2})$, which do not lie in the image.

The nicest way of phrasing it is the following. Let $\mathcal H$ be the category of Hilbert spaces with unitary maps between them. For each locally compact group $G$, one can define a functor $$Rep_G : {\mathcal H} \to Top$$ with $Rep_G(H) = \hom(G,U(H))$, where the space of homomorphisms is endowed with the compact-open topology (with respect to the strong operator topology on $U(H)$). Obviously, the functor $Rep_G$ is compatible with sums and tensor products of Hilbert spaces. Note that $Rep_{\mathbb Z}(H)=U(H).$

Consider now any such functor $F: {\mathcal H} \to Top$ and set $$D(F) = Nat_{\otimes,\oplus}(F,Rep_{\mathbb Z}),$$ i.e. all natural transformations of functors which are compatible with the tensor-product and the sum. $D(F)$ is a group since $Rep_{\mathbb Z}(H)=U(H)$ is a group for each Hilbert space $H$. It is also a topological group in a natural way.

Now, there is a natural map $\iota_G : G \to D(Rep_G)$ which is given by $\iota(g)(\pi) = \pi(g)$, where $\pi \in hom(G,U(H))$. So just as in the case of Pntrjagin duality, there is a natural bi-dual. A bit of work (relying on results of Takesaki and Gel'fand-Raikov (which you have mentioned)) shows that $\iota_G$ is a topological isomorphism for all locally compact topological groups.

I studied the analogous question which arises if one restricts everything to the category of finite-dimensional Hilbert spaces (see here). This sometimes goes under the name Chu duality, but is not so extensively studied. Everything works for locally compact abelian groups and compact groups by Pontrjagins result and the Tannaka-Krein theorem. However, for finitely generated discrete groups, interesting things happen. First of all, it is trivial to observe that the analogous map

$$\iota_G : G \to D_{fin}(Rep_G)$$

is injective if and only $G$ is maximally almost periodic (by a result of Mal'cev iff $G$ is residually finite). Moreover, and this is more difficult, $\iota_G$ is an isomorphism if and only if $G$ is virtually abelian. In particular, for $G={\mathbb F_2}$, the map $\iota_{\mathbb F_2}$ from $\mathbb F_2$ to $D_{fin}(Rep_{\mathbb F_2})$ is not surjective. This is a bit surprising as there are no natural candidates of elements in $D_{fin}(Rep_{\mathbb F_2})$, which do not lie in the image.