Unitary Representations of Unitary Groups

Abstract

In this paper we review and streamline some results of Kirillov, Olshanski and Pickrell on unitary representations of the unitary group $\mathop{\mathrm{U}}\nolimits (\mathcal{H})$ of a real, complex or quaternionic separable Hilbert space and the subgroup $\mathop{\mathrm{U}}\nolimits _{\infty }(\mathcal{H})$, consisting of those unitary operators g for which g −1 is compact. The Kirillov–Olshanski theorem on the continuous unitary representations of the identity component $\mathop{\mathrm{U}}\nolimits _{\infty }(\mathcal{H})_{0}$ asserts that they are direct sums of irreducible ones which can be realized in finite tensor products of a suitable complex Hilbert space. This is proved and generalized to inseparable spaces. These results are carried over to the full unitary group by Pickrell’s theorem, asserting that the separable unitary representations of $\mathop{\mathrm{U}}\nolimits (\mathcal{H})$, for a separable Hilbert space $\mathcal{H}$, are uniquely determined by their restriction to $\mathop{\mathrm{U}}\nolimits _{\infty }(\mathcal{H})_{0}$. For the 10 classical infinite rank symmetric pairs (G, K) of non-unitary type, such as $(\mathop{\mathrm{GL}}\nolimits (\mathcal{H}),\mathop{\mathrm{U}}\nolimits (\mathcal{H}))$, we also show that all separable unitary representations are trivial.

Appendices

A Positive Definite Functions

In this appendix we recall some results and definitions concerning operator-valued positive definite functions.

Definition A.1.

Let $\mathcal{A}$ be a C ^∗-algebra and X be a set. A map $Q: X \times X \rightarrow \mathcal{A}$ is called a positive definite kernel if, for any finite sequence $(x_{1},\ldots,x_{n}) \in X^{n},$ the matrix $Q(x_{i},x_{j})_{i,j=1,\ldots,n} \in M(n,\mathcal{A})$ is a positive element.

For $\mathcal{A} = B(V )$, V a complex Hilbert space, this means that, for $v_{1},\ldots,v_{n} \in V$, we always have $\sum _{i,j=1}^{n}\langle Q(x_{i},x_{j})v_{j},v_{i}\rangle \geq 0$.

Definition A.2.

Let $\mathcal{K}$ be a Hilbert space, G be a group, and U ⊆ G be a subset. A function $\varphi: UU^{-1} \rightarrow B(\mathcal{K})$ is said to be positive definite if the kernel

$$\displaystyle{Q_{\varphi }: U \times U \rightarrow B(\mathcal{K}),\quad (x,y)\mapsto \varphi (xy^{-1})}$$

is positive definite. For U = G we obtain the usual concept of a positive definite function on G.

Remark A.3 (Vector-Valued GNS-Construction).

We briefly recall the bridge between positive definite functions and unitary representations.

• (a) If $(\pi,\mathcal{H})$ is a unitary representation of G, $V \subseteq \mathcal{H}$ a closed subspace and $P_{V }: \mathcal{H}\rightarrow V$ the orthogonal projection on V, then $\pi _{V }(g):= P_{V }\pi (g)P_{V }^{{\ast}}$ is a B(V )-valued positive definite function with π _V (1) = 1.

• (b) If, conversely, $\varphi: G \rightarrow B(V )$ is positive definite with $\varphi (\mathbf{1}) = \mathbf{1}$, then there exists a unique Hilbert subspace $\mathcal{H}_{\varphi }$ of the space V ^G of V -valued function on G for which the evaluation maps $K_{g}: \mathcal{H}_{\varphi }\rightarrow V,f\mapsto f(g)$ are continuous and satisfy $K_{g}K_{h}^{{\ast}} =\varphi (gh^{-1})$ for g, h ∈ G [Ne00, Thm. I.1.4]. Then right translation by elements of G defines a unitary representation $(\pi _{\varphi }(g)f)(x) = f(xg)$ on this space with $K_{xg} = K_{x} \circ \pi (g)$. It is called the GNS-representation associated to ρ. Now $K_{1}^{{\ast}}: V \rightarrow \mathcal{H}_{\varphi }$ is an isometric embedding, so that we may identify V with a closed subspace of $\mathcal{H}_{\varphi }$ and K ₁ with the orthogonal projection to V. This leads to $\varphi (g) = K_{g}K_{1}^{{\ast}} = K_{\mathbf{1}}\pi (g)K_{\mathbf{1}}^{{\ast}}$, so that every positive definite function is of the form π _V . The construction also implies that $V \mathop{\cong}K_{1}^{{\ast}}(V )$ is G-cyclic in $\mathcal{H}_{\varphi }$.

For the following theorem, we simply note that all Banach–Lie groups are in particular Fréchet–BCH–Lie groups.

Theorem A.4.

Let G be a connected Fréchet–BCH–Lie group and U ⊆ G an open connected 1 -neighborhood for which the natural homomorphism $\pi _{1}(U,\mathbf{1}) \rightarrow \pi _{1}(G)$ is surjective. If $\mathcal{K}$ is Hilbert space and $\varphi: UU^{-1} \rightarrow B(\mathcal{K})$ an analytic positive definite function, then there exists a unique analytic positive definite function $\tilde{\varphi }: G \rightarrow B(\mathcal{K})$ extending  $\varphi$ .

Proof.

Let $q_{G}: \tilde{G} \rightarrow G$ be the universal covering morphism. The assumption that $\pi _{1}(U) \rightarrow \pi _{1}(G)$ is surjective implies that $\tilde{U}:= q_{G}^{-1}(U)$ is connected. Now $\tilde{\varphi }:=\varphi \circ q_{G}: \tilde{U}\tilde{U}^{-1} \rightarrow B(\mathcal{K})$ is an analytic positive definite function, hence extends by [Ne12, Thm. A.7] to an analytic positive definite function $\tilde{\varphi }$ on $\tilde{G}$. The restriction of $\tilde{\varphi }$ to $\tilde{U}$ is constant on the fibers of q _G , which are of the form $g\ker (q_{G})$. Using analyticity, we conclude that $\tilde{\varphi }(gd) =\tilde{\varphi } (g)$ holds for all $g \in \tilde{ G}$ and $d \in \ker (q_{G})$. Therefore $\tilde{\varphi }$ factors through an analytic function $\varphi: G \rightarrow B(U)$ which is obviously positive definite. □ 

Theorem A.5.

Let G be a connected analytic Fréchet–Lie group. Then a positive definite function $\varphi: G \rightarrow B(V )$ which is analytic in an open identity neighborhood is analytic.

Proof.

Since $\varphi$ is positive definite, there exists a Hilbert space $\mathcal{H}$ and a $Q: G \rightarrow B(\mathcal{H},V )$ with $\varphi (gh^{-1}) = Q_{g}Q_{h}^{{\ast}}$ for g, h ∈ G. Then the analyticity of the function $\varphi$ in an open identity neighborhood of G implies that the kernel $(g,h)\mapsto Q_{g}Q_{h}^{{\ast}}$ is analytic on a neighborhood of the diagonal $\varDelta _{G} \subseteq G \times G$. Therefore Q is analytic by [Ne12, Thm. A.3], and this implies that $\varphi (g) = Q_{g}Q_{\mathbf{1}}^{{\ast}}$ is analytic. □ 

The following proposition describes a natural source of operator-valued positive definite functions.

Proposition A.6.

Let $(\pi,\mathcal{H})$ be a unitary representation of the group G and H ⊆ G be a subgroup. Let $V \subseteq \mathcal{H}$ be an isotypic H-subspace generating the G-module $\mathcal{H}$ and $P_{V } \in B(\mathcal{H})$ be the orthogonal projection onto V. Then V is invariant under the commutant $\pi (G)' = B_{G}(\mathcal{H})$ and the map

$$\displaystyle{\gamma: B_{G}(\mathcal{H}) \rightarrow B_{H}(V ),\quad \gamma (A) = P_{V }AP_{V }}$$

is an injective morphism of von Neumann algebras whose range is the commutant of the image of the operator-valued positive definite function

$$\displaystyle{\pi _{V }: G \rightarrow B(V ),\quad \pi _{V }(g):= P_{V }\pi (g)P_{V }.}$$

In particular, if the H-representation on V is irreducible, then so is π.

Proof.

That γ is injective follows from the assumption that V generates $\mathcal{H}$ under G. If the representation (ρ, V ) of H is irreducible, then $\mathop{\mathrm{im}}\nolimits (\gamma ) \subseteq \mathbb{C}\mathbf{1}$ implies that $\pi (G)' = \mathbb{C}\mathbf{1}$, so that π is irreducible.

We now determine the range of γ. For any $A \in B_{G}(\mathcal{H})$, we have

$$\displaystyle{P_{V }\pi (g)P_{V }P_{V }AP_{V } = P_{V }\pi (g)AP_{V } = P_{V }A\pi (g)P_{V } = P_{V }AP_{V }P_{V }\pi (g)P_{V },}$$

i.e., γ(A) = P _V AP _V commutes with π _V (G). Since γ is a morphism of von Neumann algebras, its range is also a von Neumann algebra of V commuting with π _V (G). If, conversely, an orthogonal projection $Q = Q^{{\ast}} = Q^{2} \in B_{K}(V )$ commutes with π _V (G), then

$$\displaystyle{P_{V }\pi (G)QV = P_{V }\pi (G)P_{V }QV = QP_{V }\pi (G)P_{V }V \subseteq QV }$$

implies that the closed G-invariant subspace $\mathcal{H}_{Q} \subseteq \mathcal{H}$ generated by QV satisfies $P_{V }\mathcal{H}_{Q} \subseteq QV$, and therefore $\mathcal{H}_{Q} \cap V = QV$. For the orthogonal projection $\tilde{Q} \in B(\mathcal{H})$ onto $\mathcal{H}_{Q}$, which is contained in $B_{G}(\mathcal{H})$, this means that $\tilde{Q}\vert _{V } = Q$. This shows that $\mathop{\mathrm{im}}\nolimits (\gamma ) =\pi _{V }(G)'.$ □ 

Remark A.7.

The preceding proposition is particularly useful if we have specific information on the set π _V (G). As $\pi _{V }(h_{1}gh_{2}) =\rho (h_{1})\pi _{V }(g)\rho (h_{2})$, it is determined by the values of π _V on representatives of the H-double cosets in G.

• (a) In the context of the lowest K-type (ρ, V ) of a unitary highest weight representation (cf. [Ne00]), we can expect that $\pi _{V }(G) \subseteq \rho _{\mathbb{C}}(K_{\mathbb{C}})$ (by Harish–Chandra decomposition), so that $\pi _{V }(G)' =\rho _{\mathbb{C}}(K_{\mathbb{C}})' =\rho (K)'$ and γ is surjective.

• (b) In the context of Sect. 3 and [Ol78], the representation (ρ, V ) of H extends to a representation $\tilde{\rho }$ of a semigroup S ⊇ H and we obtain $\pi _{V }(G)' =\tilde{\rho } (S)'$.

In both situations we have a certain induction procedure from representations of K and S, respectively, to G-representations which preserves the commutant but which need not be defined for every representation of K, resp., S.

Lemma A.8 ( [NO13, Lemma C.3]).

Let (S,∗) be a unital involutive semigroup and $\varphi: S \rightarrow B(\mathcal{F})$ be a positive definite function with $\varphi (\mathbf{1}) = \mathbf{1}$ . We write $(\pi _{\varphi },\mathcal{H}_{\varphi })$ for the representation on the corresponding reproducing kernel Hilbert space $\mathcal{H}_{\varphi }\subseteq \mathcal{F}^{S}$ by $(\pi _{\varphi }(s)f)(t):= f(ts)$ . Then the inclusion

$$\displaystyle{\iota: \mathcal{F}\rightarrow \mathcal{H}_{\varphi },\quad \iota (v)(s):=\varphi (s)v}$$

is surjective if and only if $\varphi$ is multiplicative, i.e., a representation.

Remark A.9.

The preceding lemma can also be expressed without referring to positive definite functions and the corresponding reproducing kernel space. In this context it asserts the following. Let $\pi: S \rightarrow B(\mathcal{H})$ be a ∗-representation of a unital involutive semigroup (S, ∗), $\mathcal{F}\subseteq \mathcal{H}$ a closed cyclic subspace and $P: \mathcal{H}\rightarrow \mathcal{F}$ the orthogonal projection. Then the function

$$\displaystyle{\varphi: S \rightarrow B(\mathcal{F}),\quad \varphi (s):= P\pi (s)P^{{\ast}}}$$

is multiplicative if and only if $\mathcal{F} = \mathcal{H}$.

B C ^∗-Methods for Direct Limit Groups

In this appendix we explain how to apply C ^∗-techniques to obtain direct integral decompositions of unitary representations of direct limit groups.

We recall that for a C ^∗-algebra $\mathcal{A}$, its multiplier algebra $M(\mathcal{A})$ is a C ^∗-algebra containing $\mathcal{A}$ as an ideal, and in every faithful representation $\mathcal{A}\hookrightarrow B(\mathcal{H})$, it is given by

$$\displaystyle{M(\mathcal{A}) =\{ M \in B(\mathcal{H}): M\mathcal{A} + \mathcal{A}M \subseteq \mathcal{A}\}.}$$

Let $G = \lim _{\longrightarrow }\ G_{n}$ be a direct limit of locally compact groups and $\alpha _{n}: G_{n} \rightarrow G_{n+1}$ denote the connecting maps. We assume that these maps are closed embeddings. Then we have natural homomorphisms

$$\displaystyle{\beta _{n}: L^{1}(G_{ n}) \rightarrow M(L^{1}(G_{ n+1}))}$$

of Banach algebras, and since the action of G _n on $L^{1}(G_{n+1})$ is continuous, β _n is nondegenerate in the sense that $\beta (L^{1}(G_{n})) \cdot L^{1}(G_{n+1})$ is dense in $L^{1}(G_{n+1})$. On the level of C ^∗-algebras we likewise obtain morphisms

$$\displaystyle{\beta _{n}: C^{{\ast}}(G_{ n}) \rightarrow M(C^{{\ast}}(G_{ n+1})).}$$

A state of G ( = normalized continuous positive definite function) now corresponds to a sequence $(\varphi _{n})$ of states of the groups G _n with $\alpha _{n}^{{\ast}}\varphi _{n+1}$=$\varphi _{n}$ for every $n \in \mathbb{N}$. Passing to the C ^∗-algebras C ^∗(G _n ), we can view these functions also as states of the C ^∗-algebras. Then the compatibility condition is that the canonical extension $\tilde{\varphi }_{n}$ of $\varphi _{n}$ to the multiplier algebra satisfies

$$\displaystyle{\beta _{n}^{{\ast}}\tilde{\varphi }_{ n+1} =\varphi _{n}.}$$

Remark B.1.

The $\ell^{1}$-direct sum $\mathcal{L}:= \oplus ^{1}L^{1}(G_{n})$ carries the structure of a Banach-∗-algebra (cf. [SV75]). Every unitary representation $(\pi,\mathcal{H})$ of G defines a sequence of nondegenerate representations $\pi _{n}: L^{1}(G_{n}) \rightarrow B(\mathcal{H}_{n})$ which are compatible in the sense that

$$\displaystyle{\pi _{n} =\alpha _{ n}^{{\ast}}\tilde{\pi }_{ n+1}.}$$

Conversely, every such sequence of representations on a Hilbert space $\mathcal{H}$ leads to a sequence $\rho _{n}: G_{n} \rightarrow \mathop{\mathrm{U}}\nolimits (\mathcal{H})$ of continuous unitary representations, which are uniquely determined by

$$\displaystyle{\rho _{n}(g) =\tilde{\pi } _{n}(\eta _{G_{n}}(g)),}$$

where $\eta _{G_{n}}: G_{n} \rightarrow M(L^{1}(G_{n}))$ denotes the canonical action by left multipliers. For $f \in L^{1}(G_{n})$ and $h \in L^{1}(G_{n+1})$ we then have

$$\displaystyle\begin{array}{rcl} & & \rho _{n+1}(\alpha _{n}(g))\pi _{n}(h)\pi _{n+1}(f) =\rho _{n+1}(\alpha _{n}(g))\pi _{n+1}(\beta _{n}(h)f) =\pi _{n+1}(\alpha _{n}(g)\beta _{n}(h)f) {}\\ & & =\pi _{n+1}(\beta _{n}(g {\ast} h)f) =\pi _{n}(g {\ast} h)\pi _{n+1}(f) =\rho _{n}(g)\pi _{n}(h)\pi _{n+1}(f), {}\\ \end{array}$$

which leads to

$$\displaystyle{\rho _{n+1} \circ \alpha _{n} =\rho _{n}.}$$

Therefore the sequence (ρ _n ) is coherent and thus defines a unitary representation of G on $\mathcal{H}$. We conclude that the continuous unitary representations of G are in one-to-one correspondence with the coherent sequences of nondegenerate representations (π _n ) of the Banach-∗-algebras $L^{1}(G_{n})$ (cf. [SV75, p. 60]).

Note that the nondegeneracy condition on the sequence (β _n ) is much stronger than the nondegeneracy condition on the corresponding representation of the algebra $\mathcal{L}$. The group G _n+1 does not act by multipliers on $L^{1}(G_{n})$, so that there is no multiplier action of G on $\mathcal{L}$. However, we have a sufficiently strong structure to apply C ^∗-techniques to unitary representations of G.

Theorem B.2.

Let $\mathcal{A}$ be a separable C ^∗ -algebra and $\pi: \mathcal{A}\rightarrow \mathcal{D}$ a homomorphism into the algebra $\mathcal{D}$ of decomposable operators on a direct integral $\mathcal{H} =\int _{ X}^{\oplus }\mathcal{H}_{x},\,d\mu (x)$ . Then there exists for each x ∈ X a representation $(\pi _{x},\mathcal{H}_{x})$ of $\mathcal{A}$ such that $\pi \mathop{\cong}\int _{X}^{\oplus }\pi _{x}\,d\mu (x)$ .

If π is nondegenerate and $\mathcal{H}$ is separable, then almost all the representations π _x are nondegenerate.

Proof.

The first part is [Dix64, Lemma 8.3.1] (see also [Ke78]). Suppose that π is nondegenerate and let $(E_{n})_{n\in \mathbb{N}}$ be an approximate identity on $\mathcal{A}$. Then π(E _n ) → 1 holds strongly in $\mathcal{H}$ and [Dix69, Ch. II, no. 2.3, Prop. 4] implies the existence of a subsequence $(n_{k})_{k\in \mathbb{N}}$ such that $\pi _{x}(E_{n_{k}}) \rightarrow \mathbf{1}$ holds strongly for almost every x ∈ X. For any such x, the representation π _x is non-degenerate. □ 

Theorem B.3.

Let $G = \lim _{\longrightarrow }\ G_{n}$ be a direct limit of separable locally compact groups with closed embeddings $G_{n}\hookrightarrow G_{n+1}$ and $(\pi,\mathcal{H})$ be a continuous separable unitary representation. For any maximal abelian subalgebra $\mathcal{A}\subseteq \pi (G)'$ , we then obtain a direct integral decomposition $\pi \mathop{\cong}\int _{X}^{\oplus }\pi _{x}\,d\mu (x)$ into continuous unitary representations of G in which $\mathcal{A}$ acts by multiplication operators.

Proof.

According to the classification of commutative W ^∗-algebras, we have $\mathcal{A}\mathop{\cong}L^{\infty }(X,\mu )$ for a localizable measure space $(X,\mathfrak{S},\mu )$ [Sa71, Prop. 1.18.1]. We therefore obtain a direct integral decomposition of the corresponding Hilbert space $\mathcal{H}$. To obtain a corresponding direct integral decomposition of the representation of G, we consider the C ^∗-algebra $\mathcal{B}$ generated by the subalgebras $\mathcal{B}_{n}$ which are generated by the image of the integrated representations $L^{1}(G_{n}) \rightarrow B(\mathcal{H})$. Then each $\mathcal{B}_{n}$ is separable and therefore $\mathcal{B}$ is also separable. Hence Theorem B.2 leads to nondegenerate representations $(\pi _{x},\mathcal{H}_{x})$ of $\mathcal{B}$ whose restriction to every $\mathcal{B}_{n}$ is nondegenerate.

In [SV75], the $\ell^{1}$-direct sum $\mathcal{L}:= \oplus _{n\in \mathbb{N}}^{1}L^{1}(G_{n})$ is used as a replacement for the group algebra. From the representation $\pi: \mathcal{L}\rightarrow \mathcal{B}$ we obtain a representation ρ _x of this Banach-∗-algebra whose restrictions to the subalgebras $L^{1}(G_{n})$ are non-degenerate. Now the argument in [SV75, p. 60] (see also Remark B.1 above) implies that the corresponding continuous unitary representations of the subgroups G _n combine to a continuous unitary representation $(\rho _{x},\mathcal{H}_{x})$ of G. □ 

Remark B.4.

Let $(\pi,\mathcal{H})$ be a continuous unitary representation of the direct limit $G = \lim _{\longrightarrow }\ G_{n}$ of locally compact groups. Let $\mathcal{A}_{n}:=\pi _{n}(C^{{\ast}}(G_{n}))$ and write $\mathcal{A}:=\langle \mathcal{A}_{n}: n \in \mathbb{N}\rangle _{C^{{\ast}}}$ for the C ^∗-algebra generated by the $\mathcal{A}_{n}$. Then $\mathcal{A}'' =\pi (G)''$ follows immediately from $\mathcal{A}_{n}'' =\pi _{n}(G_{n})''$ for each n.

From the nondegeneracy of the multiplier action of C ^∗(G _n ) on C ^∗(G _n+1) it follows that

$$\displaystyle{C^{{\ast}}(G_{ n})C^{{\ast}}(G_{ n+1}) = C^{{\ast}}(G_{ n+1}),}$$

which leads to

$$\displaystyle{\mathcal{A}_{n}\mathcal{A}_{n+1} = \mathcal{A}_{n+1}.}$$

We have a decreasing sequence of closed-∗-ideals

$$\displaystyle{\mathcal{I}_{n}:= \overline{\sum _{k\geq n}\mathcal{A}_{k}} \subseteq \mathcal{A}}$$

such that G _n acts continuously by multipliers on $\mathcal{I}_{n}$. A representation of $\mathcal{I}_{n}$ is non-degenerate if and only if its restriction to $\mathcal{A}_{n}$ is nondegenerate because $\mathcal{A}_{n}\mathcal{I}_{n} = \mathcal{I}_{n}$.

If a representation $(\rho,\mathcal{K})$ of $\mathcal{A}$ is nondegenerate on all these ideals, then it is non-degenerate on every $\mathcal{A}_{n}$, hence defines a continuous unitary representation of G.

Remark B.5.

Theorem B.3 implies in particular the validity of the disintegration arguments in [Ol78, Thm. 3.6] and [Pi90, Prop. 2,4]. In [Ol84, Lemma 2.6] one also finds a very brief argument concerning the disintegration of “holomorphic” representations, namely that all the constituents are again “holomorphic”. We think that this is not obvious and requires additional arguments.