A note on induced representations of groupoid C*-algebras

In this post, which is based on a seminar series on groupoids held in Leiden in late 2022, I collect some of the basic properties of induced representations of groupoid C*-algebras. Virtually everything in this post can be found in chapter 5 of the book of Williams (cited below) with more examples, fewer mistakes and better english. The main focus will be the full groupoid C*-algebra, and we barely mention the definition of unitary representations of groupoids. In the sequel, unless stated otherwise, ${G}$ will denote a second countable locally compact Hausdorff groupoid with a Haar system ${\lambda}$ .

1. Induced representations finite groups

Given a subgroup of a finite group ${H\subset G}$ , the algebra ${{\mathbb C} G}$ has a natural right ${{\mathbb C} H}$ -action and a left ${{\mathbb C} G}$ action, both by convolution. Any unitary representation ${u: H\rightarrow B(V_u)}$ can “induce” a unitary representation of ${G}$ by the following three step process

Extend ${u}$ to a representation ${\pi_u}$ of ${{\mathbb C} H}$ (given by the “integrated form” ${\pi_u(f) = \int_G f(g)u(g)d\lambda(g)}$ for all ${f\in C_c(G)}$ )
Extend ${\pi_u}$ to a representation of ${G}$ on ${{\mathbb C} G\otimes_{{\mathbb C} H} V_u}$ , endowed with the inner product ${\langle f\otimes v, f'\otimes v' \rangle := \langle \pi_u(f^*\star f)v , v' \rangle}$ given by left multiplication of ${{\mathbb C} G}$ on ${{\mathbb C} G}$ .
restrict this representation to ${G}$ .

This is not the only way to induce representations from a subgroup to ${G}$ . Parabolic inductions from parabolic subgroups of reductive algebraic groups are one example, but also one could very well have chosen any finite vector space with a suitable ${{\mathbb C} G - {\mathbb C} H}$ -bimodule structure in place of ${{\mathbb C} G}$ in the above process and got a finite representation of ${G}$ (see the talk on “correspondences” below).

The choice of ${{\mathbb C} G}$ has many advantages though. To list a few – we know what the induced rep resentations look like, we have a characterization of which of them are irreducible (by Machey’s machinery) and we know that this particular choice of bimodule makes induction a functor adjoint to the very natural “restriction” functor which sends a representation ${\pi}$ of ${G}$ to its restriction to ${H}$ (Frobenius reciprocity theorem).

2. Unitary representations of groupoids

For groupoids however, the notion of unitary representations is somewhat convoluted at first glance. A unitary representation of a groupoid ${G}$ is defined to be a triple ${(\mu, G^{(0)}\star H, L )}$ where

${\mu}$ is a quasi-invariant measure on ${G^{(0)}}$ , which is to say ${\lambda_u\times \mu}$ and ${\lambda^u\times \mu}$ are equivalent measures on ${G}$ , where ${\lambda^u}$ ( ${u\in G^{(0)}}$ ) is the Haar system of ${G}$ (recall that ${supp(\lambda^{u})= G^u =\{ \gamma \in G~|~ r(\gamma) = u\}}$ ) and ${\lambda_u}$ is the pushforward of ${\lambda^u}$ under inversion (so we have ${supp \lambda_u = G_u}$ ).
${ G^{(0)}\star H}$ is a Borel bundle, which is a type of measurable bundle of Hilbert spaces over ${G^{(0)}}$ . (see def. 3.32 for the precise definition)
is a Borel morphism of groupoids

$L : G\rightarrow Iso(G^{(0)}\star H)\quad L(\gamma) = (r(\gamma), L_\gamma, s(\gamma))$
where is the isomorphism groupoid of the Borel bundle, given by

$Iso(G^{(0)}\star H) = \{(x, V, y) ~|~ V: H_x \rightarrow H_y \text{ is unitary}\}.$

Just like for ordinary groups, any representation of a groupoid C*-algebra can be written as the integrated form of a unitary representation (see the Integration and Disintegration theorems of Renault). Similar statements also hold for crossed products by groupoid actions.

3. Induced representations of locally compact groups

Let ${H\subset G}$ be a closed subgroup of a locally compact group ${G}$ assumed (for ease of notation) to be unimodular. The general case can be found in for instance . The space $C_c(G)$ can be equipped with a right ${C_c(H)}$ -pre-Hilbert module structure with respect to the inner product

$\langle f, g\rangle_H(h) = \int_G \overline{f(s^{-1})}g(s^{-1}h)ds$

The Hilbert completion with respect to this inner product induces a right Hilbert ${C^\star(H)}$ -module on which ${C^\star(G)}$ -acts on the left by adjointable operators (again simply by convolution). Denote this completion by ${X_H^G}$ .

Any representation of ${\pi: C^*(H) \rightarrow B(V_\pi)}$ now induces a representation

$Ind_H^G\pi : C^*(G)\rightarrow X_H^G\otimes_\pi V_\pi \qquad$

where I have used the notation of “internal tensor products” of Hilber C*-modules, which means ${X_H^G\otimes_\pi V_\pi}$ is a Hilbert space completion of

$C_c(G)\otimes V_\pi$

with respect to the (possibly degenerate) inner product

$\langle f\otimes v, f'\otimes v'\rangle = \langle \pi(\langle f, f'\rangle_H) v, v' \rangle_\pi$

with ${\langle -,- \rangle_\pi}$ the inner product on ${V_\pi}$ . Explicitly $Ind_H^G\pi$ acts on representatives $f\otimes v\in C_c(G)\otimes V_\pi$ by

$Ind_H^G\pi(f\)(f\otimes v) = f'\star f\otimes v.$

When ${X_H^G}$ can be chose to be an imprimitivity bimodule (meaning ${C^*(G) \simeq \mathbb{K}(X_H^G)}$ ), all unitary representations of ${G}$ can be induced from those of ${H}$ . The imprimitivity theorem tells us which unitary representations of ${G}$ are induced from ${H}$ . The short version of this theorem goes as follows: A unitary representation ${u:G \rightarrow U(B(H_\sigma))}$ is induced from a unitary representation of ${H}$ if and only if there is a non-degenerate representation ${\pi: C(G/H)\rightarrow B(H_\sigma)}$ such that ${(\pi, \sigma)}$ is a covariant representation of the dynamical system ${(C(G/H), G, lt)}$ , meaning

$\pi(lt_g(f)) = u_g \pi(f)u_g^\star$

with ${lt_g(f)(x) = f(g^{-1}x)}$ .

4. Induced representations of Groupoids

Let ${H \subset G}$ be a closed subgroupoid of ${G}$ with Haar systems ${(\alpha^u)_{u\in H^0}}$ and ${(\lambda^u)_{u\in G^0}}$ respectively. Again, the role of ${{\mathbb C} G}$ is replaced by a right ${C^\star(H)}$ -module. To construct this space, we start with the (closed) subset

$G_{H^{(0)}} = s^{-1}(H^0) = \{ \gamma \in G ~|~ s(\gamma ) \in H^{(0)}\}$

For our groupoid ${G}$ however, we will replace ${C_c(G)}$ with ${C_c(G_{H^0})}$ where

$G_{H^0} = s^{-1}(H^0) = \{ \gamma \in G ~|~ s(\gamma)\in H^0\}.$

Note that if ${G}$ is a group, then ${G^0 }$ is a single point, hence ${G_{H^0} = G}$ . This is a closed subspace of ${G}$ containing ${H}$ . The function algebra ${C_c(G_{H^0})}$ has a right ${C_c(H)}$ action, a ${C_c(H)}$ -linear inner product and a left ${C_c(G)}$ -action given for ${f\in C_c(G)}$ , ${\phi, \psi \in C_c(G_{H^0})}$ , ${f' \in C_c(H)}$ , ${h\in H}$ by

$\begin{align*} (f\cdot \phi)(\xi) & = \int_G f(\xi) \phi(\gamma^{-1} \xi) d\lambda^{r(\xi)}(\gamma)\\\phi f'(\xi) &= \int_H \phi(\xi h) f'(h^{-1}) d\alpha^{s(\xi)}(h) \\\langle \phi, \psi\rangle_\star (h) &= \int_G \overline{\phi(\gamma)} \psi (\gamma h) d\lambda_{r(h)}(\gamma).\\\end{align*}$

Given a representation ${L: C^\star(H) \rightarrow B(H_L)}$ , we can form the Hilbert space ${H_{indL}}$ as the Hilbert completion of the (possibly degenerate) pre-Hilbert space

$C_c(G_{H^{(0)}}) \otimes H_L$

with respect to the inner product

$(\phi\otimes h ~|~ \psi\otimes k):= \langle L(\langle \phi, \psi\rangle_\star) h, k \rangle.$

Now the induced representation

$Ind~L: C^\star(G) \rightarrow B(H_{indL})$

is determined by sending an ${f\in C_c(G)}$ to the operator acting on ${\xi\otimes v\in C_c(G_{H^0})\otimes H_L \subset H_{indL}}$ by

$(Ind_H^G L)(f)(\xi\otimes_\pi v)= f\cdot \xi \otimes_\pi v.$

In summary, other than the explicit realization of the imprimitivity correspondence ${X_H^G}$ , the process of inducing representations from closed subgroupoids runs perfectly parallel to that of ordinary group theory.

One should note that a full right Hilbert ${H}$ -module ${X}$ together with a non-degenerate left morphism

$\phi: C^*(G)\rightarrow \mathcal{L}(X)$

by adjointable operators is called a ${C^*(G)-C^\star(H)}$ –correspondence. It is customary to think of correspondences are “generalized morphisms” ${C^\star(G)\rightarrow C^\star(H)}$ , and construct a category ${\mathfrak{Corr}}$ whose objects are C*-algebras and whose morphisms are (isomorphism classes of) correspondences. This is due to the fact that many “rigidity results” and equivalences about groupoids translate only to assertions of Morita equivalence of their corresponding C*-algebras (Morita equivalences are nothing but the isomorphisms in the correspondence category). For instance the groupoid C*-algebra is independent of choice of Haar system only up to Morita equivalence. See for instance Renault’s equivalence theorem in the book of Williams.

With this setup, one can see that if ${Rep(\aa)}$ denotes the collection of unitary equivalence classes of non-degenerate representations of a C*-algebra ${\aa}$ , then

$Rep(\aa) = Hom_{\mathfrak{Corr}}(\aa, {\mathbb C})$

(a Hilbert ${{\mathbb C}}$ -module is an ordinary Hilbert space) so fixing a correspondence

$[X, \phi]\in Hom_{\mathfrak{Corr}}(\aa, \bb)$

we can define a map

$\operatorname{X-Ind}: Rep(\aa)\rightarrow Rep(\bb)$

given simply by composition in the correspondence category (which is precisely internal tensor products of Hilbert C*-modules).

For more on induction of groups/ideals/C*-algebras and the correspondence category I highly recommend chapter 2 of . For more on internal tensor products and Hilbert modules the standard reference is .

In the above we wrote ${Ind_H^G}$ or even ${Ind}$ when the groups are clear. If we use a different correspondence than the one given by ${X_H^G}$ then we will specify it by the notation $\operatorname{X-Ind}_H^G$ or simply $\operatorname{X-Ind}$ .

Since the process of groupoid induction is so similar to that of groups, it should come as no surprise that many properties translate word for word from group theory. Here we list some of the most fundamental, all of which can be found in and proofs in the cited reference therein –

4.1. Direct sums

Let

$\bigoplus L_i : C^\star(H) \rightarrow B(\bigoplus H_i)$

be a direct sum of representations then for any correspondence ${[X, \psi]: C^\star(G)\rightarrow C^*(H)}$ we have

$\operatorname{X-Ind}_H^G(\bigoplus L_i ) = \bigoplus \operatorname{X-Ind}_H^GL_i.$

4.2. Kernels

Letting ${[X, \phi] : C^\star(G)\rightarrow C^\star(H)}$ be any correspondence, one can also induce ideals from ${C^\star(H)}$ to ${C^\star(G)}$ as follows: If ${J\subset C^\star(H)}$ is an ideal then define

$\operatorname{X-Ind}_H^GJ := \{ a\in C^\star(G)~|~ \langle ax, y \rangle_H \in J, ~ \text{for all }x, y \in X\}$

This turns out to be an ideal of ${C^\star(G)}$ . With this asignment we get the formula for any representation ${\pi : C^\star(H)\rightarrow B(V_\pi)}$

$Ker Ind(\pi) = Ind(ker(\pi)).$

This will likely hold for arbitrary correspondences as well, but I haven’t found it explicitly stated, so be warned.

4.3. Induction in stages

If ${H\subset K \subset G}$ are nested closed subgroupoids of ${G}$ , and ${[Y_H^K, \phi_K^H]}$ , ${[Y_K^G, \phi_K^G]}$ are arbitrary ${C^\star(K)-C^\star(H)}$ and ${C^\star(G)-C^\star(K)}$ correspondences, then with ${[Y_H^G, \psi] = [Y_H^K, \phi_K^G]\circ [Y_K^G, \phi_K^H] = [Y_H^K\otimes_{\phi_H^K} Y_H^K, \phi_K^G\otimes 1]}$ and any representation ${\pi : C^\star(H)\rightarrow B(V_\pi)}$ we have

$\operatorname{Y_H^K-Ind_H^K}(Y_K^G-Ind_K^G \pi) = \operatorname{Y_H^G-ind}\pi$

in particular

$\operatorname{ Ind_H^K(Ind_K^G} \pi) = \operatorname{Ind_H^G} \pi.$

Let’s look at some examples of groupoid C*-algebras.

4.4. Locally closed orbits

In the case where the groupoid ${G}$ is transitive, i.e. acts transitively on its unit space ${G^0}$ , the situation is rather similar to group theory. Namely, let ${G(u): = G_u^u = \{\gamma \in G ~|~ r(\gamma) = s(\gamma) = u\}}$ be the isotropy group of ${u\in G^{(0)}}$ . Then since the action is transitive, the imprimitivity theorem gives us a Morita equivalence

$C^\star(G(u)) \sim_M C^\star(G).$

In particular, all representations of ${C^*(G)}$ are induced from an(y fixed) isotropy group. For instance, if the action of ${G}$ on ${G^0}$ has a trivial isotropy group, then ${Prim(C^*(G))}$ is a point.

The representation theory of groupoid C*-algebras hence often reduces to that of group C*-algebras.

The above theorem holds if all orbits of ${G}$ ${G^0}$ are locally closed (meaning they are open in their closure), but in this that case we cannot leave $u\in G^0$ fixed (see the comment about support of induced representations below). More precisely, we need to induce representation from distinct isotropoy groups $G(u)$ where $u$ runs over representatives of the orbit space $G^0/G$ . See .

**4.5. Regular representations and the reduced C*-norm**

The second examples are the regular representations. As opposed to the group case, there are generally more than one non-equivalent regular representation. These are given an explicit definition in chapter 1, however, they coincide with the induced representation ${L_\mu: C^\star(H) = C(G^{(0)})\rightarrow B(G^{(0)},\mu)}$ from ${H = G^{(0)}}$ given simply by left multiplication, where ${\mu}$ is a (Borel) measure on ${G^0}$ .

In this setting we have

$G_{H^{(0)}} = G \quad \text{and } \quad H = G^{(0)}$

In the case where ${\mu = \delta_u}$ is the point measure for some ${u\in G^{(0)}}$ , then we get inner products defined for ${f, f'\in C_c(G)}$ by

${\langle f, f'\rangle_\star(u) = \int_G \overline{f(\gamma)} f(\gamma u) d\lambda_{r(u)}(\gamma) = \int_G |f(\gamma)|^2 d\lambda_{u}(\gamma)}$

The Hilbert space ${H_{IndL_u}}$ is the completion of

$C_c(G)\otimes L^2(G^0, \delta_u) \simeq C_c(G)$

with respect to the inner product

$\langle f\otimes \phi, f'\otimes \phi'\rangle: = \langle L(\langle f, f'\rangle_\star)\phi, \phi' \rangle = \int_G |f(\gamma)|^2 d\lambda_{u}(\gamma) |\phi(u)|^2.$

hence we have

$H_{IndL_u} = L^2(G_u,\lambda_u)$

The induced representation now takes ${f\in C_c(G)}$ to the operator which acts on ${h \in C_c(G_u)}$ by

$Inf_{G^{(0)}}^GL_u(f)(h)(\xi) = \int_G f(\xi)\phi(\gamma^{-1}\xi) d\lambda^{r(\gamma)}(\xi)$

which is just the expression found on page 18 of for the regular representation.

Now, any representation of C*-algebra can be written as a (possibly infinite) direct sum of irreducible representations. Using the fact that induction preserves direct sums, one can conclude that if ${L: C_0(H)\rightarrow B(V)}$ is any faithful representation, then

$||Ind_G^HL(f)|| = sup_{\pi\in Rep(C_0(H))} ||Ind\pi(f)|| = ||f||_r$

This is precisely the reduced norm on ${C_c(G)}$ whose completion gives the reduced crossed product ${C_r^\star(G)}$ .

5. Induction from groupoids with closed orbits

Assume now that ${F\subsetneq G^0}$ is a closed ${G^0}$ -invariant subset of the unit space. Convince yourself that

$G|_F = G_F^F = \{ \gamma \in G ~|~ r(\gamma), s(\gamma) \in F\}$

is a subgroupoid of ${G}$ . We will show that not all representations of ${G}$ are of the form ${Ind_{G|_F}^G\pi}$ of any representation of ${C^\star(G|_F)}$ , which (by induction in stages) implies not all are of the form ${Ind_{G(u)}^G\pi}$ for ${u\in F}$ ).

To do this however, we will need some propositions. Let ${\pi}$ be as above, then we define the associated ${M}$ -representation of ${\pi}$ to be the representation ${M_\pi}$ satisfying, for every ${\phi\in C_0(H^0)}$ and ${f\in C_c(H)}$

$M(\phi)\pi(f) = \pi((\phi\circ r)\cdot f)$

where ${r}$ is the range map of ${H}$ and ${\cdot}$ is just pointwise multiplication. Explicitly we have

$M_\pi = \overline{\pi} \circ V$

where

${V: C_0(G^0)\rightarrow M(C^\star(G)) \quad \phi \mapsto (||\phi||_\infty - |\phi|^2)^{1/2}}$

and ${\overline{\pi}}$ is the extension of ${\pi}$ to ${M(C^\star(G))}$ (still non-degenerate!). Having this at our disposal, one can define the support of ${\pi}$ , to the closed subset of ${G^0}$ corresponding to the ideal ${Ker(M_\pi)}$ , that is, support of ${\pi}$ is the largest set ${C}$ for which

$\{f\in C_0(G^0)~|~ f|_C = 0\} = Ker(M_\pi).$

If ${M_\pi}$ is the usual multiplication representation on some measure space ${L^2(G^0, \mu)}$ , then the support of ${M_\pi}$ is simply the support of ${\mu}$ , hence the name.

One should really think of the support map as something that determines the essential domain of ${\pi}$ as the following proposition of shows

Theorem 1 – Let ${\pi: C^\star(G)\rightarrow B(V_\pi)}$ be any non-degenerate representation and $E\subset G^0$ a closed $G$ -invariant subset. The subset $F = supp(\pi)\subset G^0$ is also closed and $G$ -invariant and the representation ${\pi}$ factors through the (surjective) map

$j_E^G: C^\star(G)\rightarrow C^\star(G|_F)$

induced by the inclusion ${G|_E\rightarrow G}$ if and only if $F\subset E$ .

Proof

The proof is rather obvious if one believes that if ${U = G^0\backslash E}$ is the complement of ${E}$ then ${C_c(G|_U)}$ is dense in ${Ker(j_E^G)}$ . But this is always the case, as the inclusion $\iota: {C_c(G|_U) \to C_c(G)}$ sits in a short exact sequence (Theorem 5.1 of Williams)

$0\rightarrow C^\star(G|_U)\xrightarrow{\iota}C^\star(G)\xrightarrow{j_E^G}C^\star(G|_E)\rightarrow 0.$

So it suffices to check that ${C_c(G|_U) \subset ker(j_E^G)}$ . To show this, pick a ${\phi \in C_c(U)}$ such that ${(\phi\circ r) f = f}$ (i.e. a bump function which is one on all ${x\in G^0}$ where ${f(\gamma)\neq 0}$ for some ${\gamma }$ with ${r(\gamma) = 1}$ ). Then using the formula for the associated M-function above we have

$M(\phi)\pi(f) = \pi((\phi\circ r) f) = \pi(f)$

but ${M(\phi) = 0}$ since the support is ${E}$ . Hence $ker(j_E^G) \subset ker(\pi)$ and $\pi$ factors through $j_E^G$ .

Similarly one can say something about the support of an induced representation, as the following proposition from and its corollaries show

Proposition 2 – Let ${H\subset G}$ be a closed subgroupoid, ${F\subset H^0}$ closed ${H}$ -invariant subset, ${\pi: C^\star(H)\rightarrow B(V_\pi)}$ a non-degenerate representation with ${supp(\pi)\subset F}$ and assume ${E}$ is a closed ${G}$ -invariant set such that ${F\subset E \subset G^0}$ . Then as in the previous proposition ${\pi}$ factors through ${\tilde{\pi}\circ j_F^H}$ and we have

$Ind_H^G\pi = Ind_H^G(\tilde{\pi}\circ j_F^H) = Ind_{H|_F}^{G|_E}(\tilde{\pi})\circ j_{E}^G$

Note that the above proposition simply says, if a representation is induced from a representation with support ${F\subset H^0}$ then the support of the induced representation must be contained in the ${G}$ -orbit of ${\overline{GF}\subset G^0}$ . Since not all representations of $G$ have supports contained in this $G$ -orbit, it follows that not all representations of $G$ are induced from a fixed stability isotropy group.

The proof is not very glamorous so we opt to leave it out, but just mention that the map implementing the isomorphism is the restriction map

$r: C_c(G_{H^0})\rightarrow C_c(G_E^F).$

The interested reader can consult chapter 5.6 for the proof.

5.1. Amenable groups

There is a beautiful theorem which states that given a surjective morphism ${f: G\rightarrow G'}$ of locally compact groups such that ${N = Ker(f)}$ is amenable, then ${f}$ induces a morphism of the C*-algebras

$f^\star: C_r^\star(G) \rightarrow C_r^\star(G).$

In general one cannot assume the range of ${f^\star}$ is contained in ${C^\star_r(G)}$ (or even in ${C^\star(G)}$ , but only in ${M(C^\star(G))}$ ).

The reason is the following, since ${N}$ is amenable, ${C_r^\star(N) = C^\star(N)}$ which means every unitary representation lifts to ${C^\star_r(G)}$ , hence also the trivial representation ${1_N: N\rightarrow {\mathbb C}}$ . Note that we have a unitary equivalence between the representations

$Ind_N^G 1_N \sim \lambda_{G/N}$

where ${\lambda_{G/N}}$ denotes the regular representation of ${G/N}$ .

Since induced representations preserve weak containment, we have ${ker(\lambda_{N/G}) \subset ker(\lambda G)}$ hence ${f^\star}$ induces a map

$f^\star: C^\star_r(G)\rightarrow C^\star_r(G')$

since ${G' = G/N}$ .

It would be interesting to see if something similar happens in the case of groupoid C*-algebras, but this would have to be the content of a subsequent post….

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