2 The product system

Product systems over arbitrary semigroups were introduced by Fowler [Reference Fowler13], inspired by work of Arveson, and studied by several authors (see [Reference Albandik and Meyer1, Reference Carlsen, Larsen, Sims and Vittadello4, Reference Sims and Yeend26]). In this paper, we are mostly interested in product systems $\mathcal {E}$ indexed by $( \mathbb {N}^k , +)$ , associated to some representations $\rho _1,\ldots ,\rho _k$ of a compact group G. We remind the reader of some general definitions and constructions with product systems, but we restrict our attention to the Cuntz–Pimsner algebra $\mathcal {O}(\mathcal {E})$ and we mention some properties in particular cases only (see Example 2.3 for $P=\mathbb {N}^k$ ).

Definition 2.2. Given a product system $\mathcal {E}\to P$ over A and a $C^*$ -algebra B, a map $\psi :\mathcal {E}\to B$ is called a Toeplitz representation of $\mathcal {E}$ if:

• • denoting $\psi _p:=\psi |_{\mathcal {E}_p}$ , each $\psi _p:\mathcal {E}_p\to B$ is linear, $\psi _e:A\to B$ is a $*$ -homomorphism, and

  $$ \begin{align*}\psi_e(\langle x,y\rangle_p)=\psi_p(x)^*\psi_p(y)\end{align*} $$
  for all $x,y\in \mathcal {E}_p$ ; and

• • $\psi _p(x)\psi _q(y)=\psi _{pq}(xy)$ for all $p,q\in P, x\in \mathcal {E}_p, y\in \mathcal {E}_q$ .

For each $p\in P$ , we write $\psi ^{(p)}$ for the homomorphism $\mathcal {K}(\mathcal {E}_p)\to B$ obtained by extending the map $\theta _{\xi , \eta }\mapsto \psi _p(\xi )\psi _p(\eta )^*$ , where

$$ \begin{align*}\theta_{\xi, \eta}(\zeta)=\xi\langle \eta, \zeta\rangle.\end{align*} $$

The Toeplitz representation $\psi :\mathcal {E}\to B$ is Cuntz–Pimsner covariant if $\psi ^{(p)}(\phi _p(a))=\psi _e(a)$ for all $p\in P$ and all $a\in A$ such that $\phi _p(a)\in \mathcal {K}(\mathcal {E}_p)$ .

There is a $C^*$ -algebra $\mathcal {T}_A(\mathcal {E})$ called the Toeplitz algebra of $\mathcal {E}$ and a representation $i_{\mathcal {E}}:\mathcal {E}\to \mathcal {T}_A(\mathcal {E})$ which is universal in the following sense: $\mathcal {T}_A(\mathcal {E})$ is generated by $i_{\mathcal {E}}(\mathcal {E})$ and, for any representation $\psi :\mathcal {E}\to B$ , there is a homomorphism $\psi _*:\mathcal {T}_A(\mathcal {E})\to B$ such that $\psi _*\circ i_{\mathcal {E}}=\psi $ .

The Cuntz–Pimsner algebra $\mathcal {O}_A(\mathcal {E})$ of a product system $\mathcal {E}\to P$ is universal for Cuntz–Pimsner covariant representations.

There are various extra conditions on a product system $\mathcal {E}\to P$ and several other notions of covariance besides the Cuntz–Pimsner covariance from Definition 2.2, which allow one to define the Cuntz–Pimsner algebra $\mathcal {O}_A(\mathcal {E})$ or the Cuntz–Nica–Pimsner algebra $\mathcal {N}\mathcal {O}_A(\mathcal {E})$ satisfying certain properties (see [Reference Albandik and Meyer1, Reference Carlsen, Larsen, Sims and Vittadello4, Reference Dor-On and Kakariadis10, Reference Fowler13, Reference Sims and Yeend26], among others). We mention that $\mathcal {O}_A(\mathcal {E})$ (or $\mathcal {N}\mathcal {O}_A(\mathcal {E})$ ) comes with a covariant representation $j_{\mathcal {E}}:\mathcal {E}\to \mathcal {O}_A(\mathcal {E})$ and is universal in the following sense: $\mathcal {O}_A(\mathcal {E})$ is generated by $j_{\mathcal {E}}(\mathcal {E})$ and, for any covariant representation $\psi :\mathcal {E}\to B$ , there is a homomorphism $\psi _*:\mathcal {O}_A(\mathcal {E})\to B$ such that $\psi _*\circ j_{\mathcal {E}}=\psi $ . Under certain conditions, $\mathcal {O}_A(\mathcal {E})$ satisfies a gauge invariant uniqueness theorem.

Example 2.3. For a product system $\mathcal {E}\to P$ with fibers $\mathcal {E}_p$ that are nonzero finite-dimensional Hilbert spaces, and, in particular, $A=\mathcal {E}_e=\mathbb {C}$ , let us fix an orthonormal basis $\mathcal {B}_p$ in $\mathcal {E}_p$ . Then a Toeplitz representation $\psi :\mathcal {E}\to B$ gives rise to a family of isometries $\{\psi (\xi ): \xi \in \mathcal {B}_p\}_{p\in P}$ with mutually orthogonal range projections. In this case, $\mathcal {T}(\mathcal {E})=\mathcal {T}_{\mathbb {C}}(\mathcal {E})$ is generated by a collection of Cuntz–Toeplitz algebras which interact according to the multiplication maps $\mathcal {M}_{p,q}$ in $\mathcal {E}$ .

A representation $\psi :\mathcal {E}\to B$ is Cuntz–Pimsner covariant if

$$ \begin{align*} \sum_{\xi\in \mathcal{B}_p}\psi(\xi)\psi(\xi)^*=\psi(1)\end{align*} $$

for all $p\in P$ . The Cuntz–Pimsner algebra $\mathcal {O}(\mathcal {E})=\mathcal {O}_{\mathbb {C}}(\mathcal {E})$ is generated by a collection of Cuntz algebras, so it could be thought of as a multidimensional Cuntz algebra. Fowler proved in [Reference Fowler12] that if the function $p\mapsto \dim \mathcal {E}_p$ is injective, then the algebra ${\mathcal O}(\mathcal {E})$ is simple and purely infinite. For other examples of multidimensional Cuntz algebras, see [Reference Burgstaller3].

The following two definitions and two results are taken from [Reference Deaconu, Huang and Sims7]; see also [Reference Hao and Ng15, Reference Kumjian and Pask17].

Definition 2.5. An action $ \beta $ of a locally compact group $ G $ on a product system $ \mathcal {E} \to P $ over A is a family $ (\beta ^{p})_{p \in P} $ such that $ \beta ^{p} $ is an action of $ G $ on each fiber $\mathcal {E}_{p} $ compatible with the action $\alpha =\beta ^e$ on A, and, furthermore, the actions $(\beta ^p)_{p\in P}$ are compatible with the multiplication maps $\mathcal {M}_{p,q}$ in the sense that

$$ \begin{align*}\beta^{p q}_g(\mathcal{M}_{p,q}(x \otimes y)) = \mathcal{M}_{p,q}(\beta^{p}_g(x) \otimes \beta^{q}_g(y)) \end{align*} $$

for all $ g \in G $ , $ x \in \mathcal {E}_{p} $ and $ y \in \mathcal {E}_{q} $ .

Definition 2.6. If $ \beta $ is an action of $ G $ on the product system $\mathcal {E} \to P $ , we define the crossed product $\mathcal {E} \rtimes _{\beta } G $ as the product system indexed by $ P $ with fibers $ \mathcal {E}_{p} \rtimes _{\beta ^{p}} G $ , which are $ C^{\ast } $ -correspondences over $ A \rtimes _{\alpha } G $ . For $ \zeta \in C_c(G,\mathcal {E}_{p}) $ and $ \eta \in C_c(G,\mathcal {E}_{q}) $ , the product $ \zeta \eta \in C_c(G,\mathcal {E}_{p q}) $ is defined by

$$ \begin{align*}(\zeta \eta)(s) = \int_G\mathcal{M}_{p,q}(\zeta(t) \otimes \beta^{q}_t(\eta(t^{- 1} s)))\,dt. \end{align*} $$

Proposition 2.8. Suppose that a locally compact group $ G $ acts on a row-finite and faithful product system $ \mathcal {E} $ indexed by $ P = (\mathbb {N}^{k},+) $ via automorphisms $ \beta ^{p}_{g} $ . Then $ G $ acts on the Cuntz–Pimsner algebra $\mathcal {O}_{A}(\mathcal {E}) $ via automorphisms denoted by $ \gamma _{g} $ . Moreover, if $ G $ is amenable, then $ \mathcal {E} \rtimes _{\beta } G $ is row-finite and faithful, and

$$ \begin{align*}\mathcal{O}_{A}(\mathcal{E}) \rtimes_{\gamma} G \cong \mathcal{O}_{A \rtimes_{\alpha} G}(\mathcal{E} \rtimes_{\beta} G). \end{align*} $$

Now we define the product system associated to k representations of a compact group G. We limit ourselves to finite-dimensional unitary representations, even though the definition makes sense in greater generality.

Definition 2.9. Given a compact group G and k finite-dimensional unitary representations $\rho _i$ of G on Hilbert spaces $\mathcal H_i$ for $i=1,\ldots ,k$ , we construct the product system $\mathcal {E}=\mathcal {E}(\rho _1,\ldots ,\rho _k)$ indexed by the commutative monoid $(\mathbb N^k,+)$ , with fibers

$$ \begin{align*}\mathcal E_n=\mathcal{H}^n=\mathcal H_1^{\otimes n_1}\otimes\cdots\otimes \mathcal H_k^{\otimes n_k}\end{align*} $$

for $n=(n_1,\ldots ,n_k)\in \mathbb {N}^k$ ; in particular, $A=\mathcal E_0=\mathbb C$ . The multiplication maps

$$ \begin{align*}\mathcal {M}_{n,m}:\mathcal {E}_n\times \mathcal {E}_m\to \mathcal {E}_{n+m} \end{align*}$$

in $\mathcal {E}$ are defined by using the standard isomorphisms $\rho _i\otimes \rho _j\cong \rho _j\otimes \rho _i$ for all $i<j$ . The associativity in $\mathcal {E}$ follows from the fact that

$$ \begin{align*}\mathcal{M}_{n+m,p}\circ (\mathcal{M}_{n,m}\times id)=\mathcal{M}_{n,m+p}\circ ( id\times \mathcal{M}_{m,p})\end{align*} $$

as maps from $\mathcal {E}_n\times \mathcal {E}_m\times \mathcal {E}_p$ to $\mathcal {E}_{n+m+p}.$ Then $\mathcal {E}=\mathcal {E}(\rho _1,\ldots ,\rho _k)$ is called the product system of the representations $\rho _1,\ldots ,\rho _k$ .

Proposition 2.11. With notation as in Definition 2.9, assume that $d_i=\dim \mathcal {H}_i\ge 2$ . Then the Cuntz–Pimsner algebra $\mathcal {O}(\mathcal {E})$ associated to the product system $\mathcal {E}\to \mathbb {N}^k$ described above is isomorphic with the $C^*$ -algebra of a rank k graph $\Gamma $ with a single vertex and with $|\Gamma ^{\varepsilon _i}|=d_i$ . This isomorphism is equivariant for the gauge action. Moreover,

$$ \begin{align*}\mathcal{O}(\mathcal{E})\cong \mathcal O_{d_1}\otimes\cdots\otimes \mathcal O_{d_k},\end{align*} $$

where $\mathcal O_n$ is the Cuntz algebra.

Proposition 2.13. The compact group G acts on each fiber $\mathcal {E}_n$ of the product system $\mathcal E$ via the representation $\rho ^n=\rho _1^{\otimes n_1}\otimes \cdots \otimes \rho _k^{\otimes n_k}$ . This action is compatible with the multiplication maps and commutes with the gauge action of $\mathbb {T}^k$ . The crossed product $\mathcal E\rtimes G$ becomes a row-finite and faithful product system indexed by $\mathbb N^k$ over the group $C^*$ -algebra $C^*(G)$ . Moreover,

$$ \begin{align*}\mathcal{O}(\mathcal{E}) \rtimes G \cong \mathcal{O}_{C^*(G)}(\mathcal{E} \rtimes G).\end{align*} $$

Proof. Indeed, for $g\in G$ and $\xi \in \mathcal {E}_n=\mathcal {H}^n$ , we define $g\cdot \xi =\rho ^n(g)(\xi )$ , and since $\rho _i\otimes \rho _j\cong \rho _j\otimes \rho _i$ , we have $g\cdot (\xi \otimes \eta )=g\cdot \xi \otimes g\cdot \eta $ for $\xi \in \mathcal {E}_n, \eta \in \mathcal {E}_m$ . Clearly,

$$ \begin{align*}g\cdot\gamma_z(\xi)=g\cdot(z^n\xi)=z^n(g\cdot\xi)=\gamma_z(g\cdot\xi),\end{align*} $$

so the action of G commutes with the gauge action. Using Proposition 2.7, $\mathcal E\rtimes G$ becomes a product system indexed by $\mathbb N^k$ over $C^*(G)\cong \mathbb {C}\rtimes G$ with fibers $\mathcal {E}_n\rtimes G$ . The isomorphism $\mathcal {O}(\mathcal {E}) \rtimes G \cong \mathcal {O}_{C^*(G)}(\mathcal {E} \rtimes G)$ follows from Proposition 2.8.

3 The Doplicher–Roberts algebra

The Doplicher–Roberts algebras $\mathcal {O}_\rho $ , denoted by ${\mathcal O}_G$ in [Reference Doplicher and Roberts8], were introduced to construct a new duality theory for compact Lie groups G that strengthens the Tannaka–Krein duality. Here $\rho $ is the n-dimensional representation of G defined by the inclusion $G\subseteq U(n)$ in some unitary group $U(n)$ . Let ${\mathcal T}_G$ denote the representation category whose objects are tensor powers $\rho ^p=\rho ^{\otimes p}$ for $p\ge 0$ , and whose arrows are the intertwiners $Hom(\rho ^p, \rho ^q)$ . The group G acts via $\rho $ on the Cuntz algebra ${\mathcal O}_n$ and ${\mathcal O}_G={\mathcal O}_\rho $ is identified in [Reference Doplicher and Roberts8] with the fixed point algebra ${\mathcal O}_n^G$ . If $\sigma $ denotes the restriction to ${\mathcal O}_\rho $ of the canonical endomorphism of $\mathcal {O}_n$ , then ${\mathcal T}_G$ can be reconstructed from the pair $({\mathcal O}_\rho ,\sigma )$ . Subsequently, Doplicher–Roberts algebras were associated to any object $\rho $ in a strict tensor $C^*$ -category (see [Reference Doplicher and Roberts9]).

Given finite-dimensional unitary representations $\rho _1,\ldots ,\rho _k$ of a compact group G on Hilbert spaces $\mathcal H_1,\ldots , \mathcal H_k$ , we construct a Doplicher–Roberts algebra $\mathcal O_{\rho _1,\ldots ,\rho _k}$ from intertwiners

$$ \begin{align*}Hom (\rho^n, \rho^m)=\{T\in\mathcal{L}({\mathcal H}^n, {\mathcal H}^m)\;\mid \; T\rho^n(g)=\rho^m(g)T \text{ for all } g\in G\},\end{align*} $$

where, for $n=(n_1,\ldots ,n_k)\in \mathbb N^k$ , the representation $\rho ^n=\rho _1^{\otimes n_1}\otimes \cdots \otimes \rho _k^{\otimes n_k}$ acts on $\mathcal {H}^n=\mathcal H_1^{\otimes n_1}\otimes \cdots \otimes \mathcal H_k^{\otimes n_k}$ . Note that $\rho ^0=\iota $ is the trivial representation of G, acting on $\mathcal {H}^0=\mathbb {C}$ . This Doplicher–Roberts algebra is a subalgebra of $\mathcal {O}(\mathcal {E})$ for the product system $\mathcal {E}$ , as in Definition 2.9.

Lemma 3.1. Consider

$$ \begin{align*} \mathcal{A}_0=\bigcup_{m,n\in \mathbb{N}^k}\mathcal{L}(\mathcal H^n,\mathcal H^m).\end{align*} $$

Then the linear span of $\mathcal {A}_0$ becomes a $*$ -algebra $\mathcal {A}$ with appropriate multiplication and involution. This algebra has a natural $\mathbb {Z}^k$ -grading coming from a gauge action of $\mathbb {T}^k$ . Moreover, the Cuntz–Pimsner algebra $\mathcal {O}(\mathcal {E})$ of the product system $\mathcal {E}=\mathcal {E}(\rho _1,\ldots ,\rho _k)$ is equivariantly isomorphic to the $C^*$ -closure of $\mathcal {A}$ in the unique $C^*$ -norm for which the gauge action is isometric.

Proof. Recall that the Cuntz algebra $\mathcal {O}_n$ contains a canonical Hilbert space $\mathcal {H}$ of dimension n and it can be constructed as the closure of the linear span of $ \bigcup _{p,q\in \mathbb {N}}\mathcal {L}(\mathcal H^p,\mathcal H^q)$ using embeddings

$$ \begin{align*}\mathcal{L}(\mathcal{H}^p,\mathcal{H}^q)\subseteq \mathcal{L}(\mathcal{H}^{ p+1},\mathcal{H}^{ q+1}),\quad T\mapsto T\otimes I,\end{align*} $$

where $\mathcal {H}^p=\mathcal H^{\otimes p}$ and $I:\mathcal {H}\to \mathcal {H}$ is the identity map. This linear span becomes a $*$ -algebra with a multiplication given by composition and an involution (see [Reference Doplicher and Roberts8] and Proposition 2.5 in [Reference Katsoulis16]).

Similarly, for all $r\in \mathbb {N}^k$ , we consider embeddings $\mathcal {L}(\mathcal {H}^n,\mathcal {H}^m)\subseteq \mathcal {L}(\mathcal {H}^{n+r},\mathcal {H}^{m+r})$ given by $T\mapsto T\otimes I_r$ , where $I_r:{\mathcal H}^r\to {\mathcal H}^r$ is the identity map, and we endow $\mathcal {A}$ with a multiplication given by composition and an involution. More precisely, if $S\in \mathcal {L}(\mathcal {H}^n,\mathcal {H}^m)$ and $T\in \mathcal {L}(\mathcal {H}^q,\mathcal {H}^p)$ , then the product $ST$ is

$$ \begin{align*}(S\otimes I_{p\vee n-n})\circ (T\otimes I_{p\vee n-p})\in \mathcal{L}(\mathcal{H}^{q+p\vee n-p},\mathcal{H}^{m+p\vee n-n}),\end{align*} $$

where we write $p\vee n$ for the coordinatewise maximum. This multiplication is well defined in $\mathcal {A}$ and is associative. The adjoint of $T\in \mathcal {L}(\mathcal {H}^n,\mathcal {H}^m)$ is $T^*\in \mathcal {L}(\mathcal {H}^m,\mathcal {H}^n)$ .

There is a natural $\mathbb Z^k$ -grading on $\mathcal {A}$ given by the gauge action $\gamma $ of $\mathbb {T}^k$ , where, for $z=(z_1,\ldots ,z_k)\in \mathbb {T}^k$ and $T\in \mathcal {L}(\mathcal {H}^n,\mathcal {H}^m)$ , we define

$$ \begin{align*}\gamma_z(T)(\xi)=z_1^{m_1-n_1}\cdots z_k^{m_k-n_k}T(\xi).\end{align*} $$

Adapting the argument in Theorem 4.2 in [Reference Doplicher and Roberts9] for $\mathbb {Z}^k$ -graded $C^*$ -algebras, the $C^*$ -closure of $\mathcal {A}$ in the unique $C^*$ -norm for which $\gamma _z$ is isometric is well defined. The map

$$ \begin{align*}(T_1,\ldots,T_k)\mapsto T_1\otimes\cdots\otimes T_k, \end{align*} $$

where

$$ \begin{align*}T_1\otimes\cdots\otimes T_k: \mathcal H^n\to \mathcal H^m,\; (T_1\otimes\cdots\otimes T_k)(\xi_1\otimes\cdots\otimes \xi_k)=T_1(\xi_1)\otimes\cdots\otimes T_k(\xi_k)\end{align*} $$

for $T_i\in \mathcal {L}(\mathcal H_i^{n_i},\mathcal H_i^{m_i})$ for $i=1,\ldots ,k$ preserves the gauge action and it can be extended to an equivariant isomorphism from $\mathcal {O}(\mathcal {E})\cong \mathcal {O}_{d_1}\otimes \cdots \otimes \mathcal {O}_{d_k}$ to the $C^*$ -closure of $\mathcal {A}$ . Note that the closure of $ \bigcup _{n\in \mathbb {N}^k}\mathcal {L}(\mathcal H^n,\mathcal H^n)$ is isomorphic to the core $\mathcal {F}=\mathcal {O}(\mathcal {E})^{\mathbb {T}^k}$ , that is the fixed point algebra under the gauge action, which is a UHF-algebra.

To define the Doplicher–Roberts algebra $\mathcal O_{\rho _1,\ldots ,\rho _k}$ , we again identify $Hom(\rho ^n,\rho ^m)$ with a subset of $Hom(\rho ^{n+r},\rho ^{m+r})$ for each $r\in \mathbb N^k$ , via $T\mapsto T\otimes I_r$ . After this identification, it follows that the linear span ${}^0{\mathcal O}_{\rho _1,\ldots , \rho _k}$ of $ \bigcup _{m,n\in \mathbb {N}^k}Hom(\rho ^n, \rho ^m)\subseteq \mathcal {A}_0$ has a natural multiplication and involution inherited from $\mathcal {A}$ . Indeed, a computation shows that if $S\in Hom(\rho ^n, \rho ^m)$ and $T\in Hom(\rho ^q,\rho ^p)$ , then $S^*\in Hom(\rho ^m, \rho ^n)$ and

$$ \begin{align*} &((S\otimes I_{p\vee n-n})\circ (T\otimes I_{p\vee n-p}))\rho^{q+p\vee n-p}(g)\\ &\quad=\rho^{m+p\vee n-n}(g)((S\otimes I_{p\vee n-n})\circ (T\otimes I_{p\vee n-p})),\end{align*} $$

so $(S\otimes I_{p\vee n-n})\circ (T\otimes I_{p\vee n-p})\in Hom(\rho ^{q+p\vee n-p}, \rho ^{m+p\vee n-n})$ and ${}^0{\mathcal O}_{\rho _1,\ldots , \rho _k}$ is closed under these operations. Since the action of G commutes with the gauge action, there is a natural $\mathbb Z^k$ -grading of ${}^0{\mathcal O}_{\rho _1,\ldots ,\rho _k}$ given by the gauge action $\gamma $ of $\mathbb {T}^k$ on $\mathcal {A}$ .

It follows that the closure ${\mathcal O}_{\rho _1,\ldots , \rho _k}$ of ${}^0{\mathcal O}_{\rho _1,\ldots ,\rho _k}$ in $\mathcal {O}(\mathcal {E})$ is well defined, obtaining the Doplicher–Roberts algebra associated to the representations $\rho _1,\ldots ,\rho _k$ . This $C^*$ -algebra also has a $\mathbb Z^k$ -grading and a gauge action of $\mathbb {T}^k$ . By construction, ${\mathcal O}_{\rho _1,\ldots , \rho _k}\subseteq \mathcal {O}(\mathcal {E})$ .

Proof. We know from Lemma 3.1 that ${\mathcal O}(\mathcal {E})$ is isomorphic to the $C^*$ -algebra generated by the linear span of $ \mathcal {A}_0= \bigcup _{m,n\in \mathbb {N}^k}\mathcal {L}({\mathcal H}^n, {\mathcal H}^m)$ . The group G acts on $\mathcal {L}({\mathcal H}^n, {\mathcal H}^m)$ by

$$ \begin{align*}(g\cdot T)(\xi)=\rho^m(g)T(\rho^n(g^{-1})\xi)\end{align*} $$

and the fixed point set is $Hom(\rho ^n, \rho ^m)$ . Indeed, we have $g\cdot T=T$ if and only if $T\rho ^n(g)=\rho ^m(g)T$ . This action is compatible with the embeddings and the operations, so it extends to the $*$ -algebra $\mathcal {A}$ and the fixed point algebra is the linear span of $ \bigcup _{m,n\in \mathbb {N}^k}Hom(\rho ^n, \rho ^m)$ .

It follows that ${}^0{\mathcal O}_{\rho _1,\ldots ,\rho _k}\subseteq {\mathcal O}(\mathcal {E})^G$ and therefore its closure ${\mathcal O}_{\rho _1,\ldots ,\rho _k}$ is isomorphic to a subalgebra of ${\mathcal O}(\mathcal {E})^G$ . For the other inclusion, any element in ${\mathcal O}(\mathcal {E})^G$ can be approximated with an element from ${}^0{\mathcal O}_{\rho _1,\ldots ,\rho _k}$ , and hence ${\mathcal O}_{\rho _1,\ldots ,\rho _k}=\mathcal {O}(\mathcal {E})^G$ .

In the next section, we show that, in many cases, $\mathcal O_{\rho _1,\ldots ,\rho _k}$ is isomorphic to a corner of $C^*(\Lambda )$ for a rank k graph $\Lambda $ , so, in some cases, we can compute its K-theory. It would be nice to express the K-theory of $\mathcal {O}_{\rho _1,\ldots ,\rho _k}$ in terms of the maps $\pi \mapsto \pi \otimes \rho _i$ defined on the representation ring $\mathcal {R}(G)$ .

4 The rank k graphs

For convenience, we first collect some facts about higher rank graphs, introduced in [Reference Kumjian, Pask, Raeburn and Renault18]. A rank k graph or k-graph $(\Lambda , d)$ consists of a countable small category $\Lambda $ with range and source maps r and s together with a functor $d : \Lambda \rightarrow \mathbb {N}^k$ called the degree map, satisfying the factorization property: for every $\lambda \in \Lambda $ and all $m, n \in \mathbb {N}^k$ with $d( \lambda ) = m + n$ , there are unique elements $\mu , \nu \in \Lambda $ such that $\lambda = \mu \nu $ and $d( \mu ) = m$ , $d( \nu ) = n$ . For $n \in \mathbb {N}^k$ , we write $\Lambda ^n := d^{-1} (n)$ and call it the set of paths of degree n. For $\varepsilon _i=(0,\ldots ,1,\ldots ,0)$ with $1$ in position i, the elements in $\Lambda ^{\varepsilon _i}$ are called edges and the elements in $\Lambda ^0$ are called vertices.

A k-graph $\Lambda $ can be constructed from $\Lambda ^0$ and from its k-colored skeleton $\Lambda ^{\varepsilon _1}\cup \cdots \cup \Lambda ^{\varepsilon _k}$ using a complete and associative collection of commuting squares or factorization rules (see [Reference Sims25]).

The k-graph $\Lambda $ is row-finite if, for all $n\in \mathbb {N}^k$ and all $v\in \Lambda ^0$ , the set $v\Lambda ^n := \{\lambda \in \Lambda ^n : r(\lambda ) = v\}$ is finite. It has no sources if $v\Lambda ^n\neq \emptyset $ for all $v\in \Lambda ^0$ and $n\in \mathbb {N}^k$ . A k-graph $\Lambda $ is said to be irreducible (or strongly connected) if, for every $u,v\in \Lambda ^0$ , there is $\lambda \in \Lambda $ such that $u = r(\lambda )$ and $v = s(\lambda )$ .

Recall that $C^*(\Lambda )$ is the universal $C^*$ -algebra generated by a family $\{S_\lambda : \lambda \in \Lambda \}$ of partial isometries satisfying:

• • $\{S_v:v\in \Lambda ^0\}$ is a family of mutually orthogonal projections;

• • $S_{\lambda \mu }=S_\lambda S_\mu $ for all $\lambda , \mu \in \Lambda $ such that $s(\lambda ) = r(\mu )$ ;

• • $S_\lambda ^*S_\lambda = S_{s(\lambda )}$ for all $\lambda \in \Lambda $ ; and

• • for all $v\in \Lambda ^0$ and $n\in \mathbb {N}^k$ ,

  $$ \begin{align*} S_v=\sum_{\lambda\in v\Lambda^n}S_\lambda S_\lambda^*.\end{align*} $$

A k-graph $\Lambda $ is said to satisfy the aperiodicity condition if, for every vertex $v\in \Lambda ^0$ , there is an infinite path $x\in v\Lambda ^\infty $ such that $\sigma ^mx\neq \sigma ^nx$ for all $m\neq n$ in $\mathbb {N}^k$ , where $\sigma ^m:\Lambda ^\infty \to \Lambda ^\infty $ are the shift maps. We say that $\Lambda $ is cofinal if, for every $x\in \Lambda ^\infty $ and $v\in \Lambda ^0$ , there is $\lambda \in \Lambda $ and $n\in \mathbb {N}^k$ such that $s(\lambda )=x(n)$ and $r(\lambda )=v$ .

Assume that $\Lambda $ is row-finite with no sources and that it satisfies the aperiodicity condition. Then $C^*(\Lambda )$ is simple if and only if $\Lambda $ is cofinal (see Proposition 4.8 in [Reference Kumjian, Pask, Raeburn and Renault18] and Theorem 3.4 in [Reference Robertson and Sims23]).

We say that a path $\mu \in \Lambda $ is a loop with an entrance if $s(\mu )=r(\mu )$ , and there exists $\alpha \in s(\mu )\Lambda $ such that $d(\mu )\ge d(\alpha )$ and there is no $\beta \in \Lambda $ with $\mu = \alpha \beta $ . We say that every vertex connects to a loop with an entrance if, for every $v\in \Lambda ^0$ , there is a loop with an entrance $\mu \in \Lambda $ , and a path $\lambda \in \Lambda $ with $r(\lambda )=v$ and $s(\lambda )=r(\mu )=s(\mu )$ . If $\Lambda $ satisfies the aperiodicity condition and every vertex connects to a loop with an entrance, then $C^*(\Lambda )$ is purely infinite (see Proposition 4.9 in [Reference Kumjian, Pask, Raeburn and Renault18] and Proposition 8.8 in [Reference Sims24]).

Given finite-dimensional unitary representations $\rho _i$ of a compact group G on Hilbert spaces $\mathcal H_i$ for $i=1,\ldots ,k$ , we want to construct a rank k graph $\Lambda =\Lambda (\rho _1,\ldots ,\rho _k)$ . Let R be the set of equivalence classes of irreducible summands $\pi :G\to U(\mathcal {H}_\pi )$ which appear in the tensor powers $\rho ^n=\rho _1^{\otimes n_1}\otimes \cdots \otimes \rho _k^{\otimes n_k}$ for $n\in \mathbb {N}^k$ , as in [Reference Mann, Raeburn and Sutherland22]. Take $\Lambda ^0=R$ and, for each $i=1,\ldots ,k$ , consider the set of edges $\Lambda ^{\varepsilon _i}$ which are uniquely determined by the matrices $M_i$ with entries

$$ \begin{align*}M_i(w,v)=|\{e\in \Lambda^{\varepsilon_i}: s(e)=v, r(e)=w\}|=\dim Hom(v,w\otimes \rho_i),\end{align*} $$

where $v,w\in R$ . The matrices $M_i$ commute since $\rho _i\otimes \rho _j\cong \rho _j\otimes \rho _i$ and therefore

$$ \begin{align*}\dim Hom(v,w\otimes \rho_i\otimes\rho_j)=\dim Hom(v,w\otimes \rho_j\otimes\rho_i)\end{align*} $$

for all $i<j$ . This allows us to fix some bijections

$$ \begin{align*}\lambda_{ij}:\Lambda^{\varepsilon_i}\times_{\Lambda^0}\Lambda^{\varepsilon_j}\to \Lambda^{\varepsilon_j}\times_{\Lambda^0}\Lambda^{\varepsilon_i}\end{align*} $$

for all $1\le i<j\le k$ , which determine the commuting squares of $\Lambda $ . As usual,

$$ \begin{align*}\Lambda^{\varepsilon_i}\times_{\Lambda^0}\Lambda^{\varepsilon_j}=\{(e,f)\in \Lambda^{\varepsilon_i}\times \Lambda^{\varepsilon_j}: s(e)=r(f)\}.\end{align*} $$

For $k\ge 3$ , we also need to verify that $\lambda _{ij}$ can be chosen to satisfy the associativity condition, that is,

$$ \begin{align*}(id_\ell\times \lambda_{ij})(\lambda_{i\ell}\times id_j)(id_i\times \lambda_{j\ell})=(\lambda_{j\ell}\times id_i)(id_j\times \lambda_{i\ell})(\lambda_{ij}\times id_\ell)\end{align*} $$

as bijections from $\Lambda ^{\varepsilon _i}\times _{\Lambda ^0}\Lambda ^{\varepsilon _j}\times _{\Lambda ^0}\Lambda ^{\varepsilon _\ell }$ to $\Lambda ^{\varepsilon _\ell }\times _{\Lambda ^0}\Lambda ^{\varepsilon _j}\times _{\Lambda ^0}\Lambda ^{\varepsilon _i}$ for all $i<j<\ell $ .

Remark 4.3. Note that the entry $M_i(w,v)$ is just the multiplicity of the irreducible representation v in $w\otimes \rho _i$ for $i=1,\ldots ,k$ . If $\rho _i^*=\rho _i$ , then the matrices $M_i$ are symmetric since

$$ \begin{align*}\dim Hom(v, w\otimes \rho_i)=\dim Hom(\rho^*_i\otimes v,w)\end{align*} $$

which implies $M_i(w; v) = M_i(v;w)$ . Here $\rho ^*_i$ denotes the dual representation defined by $\rho _i^*(g)=\rho _i(g^{-1})^t$ and equal, in our case, to the conjugate representation $\bar {\rho _i}$ .

For G finite, these matrices are finite, and the entries $M_i(w,v)$ can be computed using the character table of G. For G infinite, the Clebsch–Gordan relations can be used to determine the numbers $M_i(w,v)$ . Since the bijections $\lambda _{ij}$ are, in general, not unique, the rank k graph $\Lambda $ is not unique, as illustrated in some examples. It is an open question how the $C^*$ -algebra $C^*(\Lambda )$ depends, in general, on the factorization rules.

To relate the Doplicher–Roberts algebra $\mathcal {O}_{\rho _1,\ldots ,\rho _k}$ to a rank k graph $\Lambda $ , we mimic the construction in [Reference Mann, Raeburn and Sutherland22]. For each edge $e\in \Lambda ^{\varepsilon _i}$ , choose an isometric intertwiner

$$ \begin{align*}T_e: \mathcal{H}_{s(e)}\to \mathcal{H}_{r(e)}\otimes \mathcal{H}_i\end{align*} $$

in such a way that

$$ \begin{align*}\mathcal{H}_\pi\otimes \mathcal{H}_i=\bigoplus_{e\in \pi\Lambda^{\varepsilon_i}}T_eT_e^*(\mathcal{H}_\pi\otimes \mathcal{H}_i)\end{align*} $$

for all $\pi \in \Lambda ^0$ , that is, the edges in $\Lambda ^{\varepsilon _i}$ ending at $\pi $ give a specific decomposition of $\mathcal {H}_\pi \otimes \mathcal {H}_i$ into irreducibles. When $\dim Hom(s(e), r(e)\otimes \rho _i)\ge 2$ , we must choose a basis of isometric intertwiners with orthogonal ranges, so, in general, $T_e$ is not unique. In fact, specific choices for the isometric intertwiners $T_e$ determine the factorization rules in $\Lambda $ and whether or not they satisfy the associativity condition.

Given $e\in \Lambda ^{\varepsilon _i}$ and $f\in \Lambda ^{\varepsilon _j}$ with $r(f)=s(e)$ , we know how to multiply $T_e\in Hom(s(e),r(e)\otimes \rho _i)$ with $T_f\in Hom(s(f),r(f)\otimes \rho _j)$ in the algebra $\mathcal {O}_{\rho _1,\ldots ,\rho _k}$ , by viewing $Hom(s(e),r(e)\otimes \rho _i)$ as a subspace of $Hom(\rho ^n,\rho ^m)$ for some $m,n$ , and similarly for $Hom(s(f),r(f)\otimes \rho _j)$ . We choose edges $e'\in \Lambda ^{\varepsilon _i}, f'\in \Lambda ^{\varepsilon _j}$ with $s(f)=s(e'), r(e)=r(f'), r(e')=s(f')$ such that $T_eT_f=T_{f'}T_{e'}$ , where $T_{f'}\in Hom(s(f'),r(f')\otimes \rho _j)$ and $T_{e'}\in Hom(s(e'),r(e')\otimes \rho _i)$ . This is possible since

$$ \begin{align*}T_eT_f=(T_e\otimes I_j)\circ T_f\in Hom(s(f),r(e)\otimes \rho_i\otimes \rho_j),\end{align*} $$

$$ \begin{align*}T_{f'}T_{e'}=(T_{f'}\otimes I_i)\circ T_{e'}\in Hom(s(e'),r(f')\otimes \rho_j\otimes \rho_i),\end{align*} $$

and $\rho _i\otimes \rho _j\cong \rho _j\otimes \rho _i$ . In this case, we declare that $ef=f'e'$ . Repeating this process, we obtain bijections $\lambda _{ij}:\Lambda ^{\varepsilon _i}\times _{\Lambda ^0}\Lambda ^{\varepsilon _j}\to \Lambda ^{\varepsilon _j}\times _{\Lambda ^0}\Lambda ^{\varepsilon _i}$ . Assuming that the associativity conditions are satisfied, we obtain a k-graph $\Lambda $ .

We write $T_{ef}=T_eT_f=T_{f'}T_{e'}=T_{f'e'}$ . A finite path $\lambda \in \Lambda ^n$ is a concatenation of edges and determines by composition a unique intertwiner

$$ \begin{align*}T_\lambda:\mathcal{H}_{s(\lambda)}\to \mathcal{H}_{r(\lambda)}\otimes\mathcal{H}^n.\end{align*} $$

Moreover, the paths $\lambda \in \Lambda ^n$ with $r(\lambda )=\iota $ , the trivial representation, provide an explicit decomposition of $\mathcal {H}^n=\mathcal {H}_1^{\otimes n_1}\otimes \cdots \otimes \mathcal {H}_k^{\otimes n_k}$ into irreducibles, and hence

$$ \begin{align*}\mathcal{H}^n=\bigoplus_{\lambda\in\iota \Lambda^n}T_\lambda T_\lambda^*(\mathcal{H}^n).\end{align*} $$

Proposition 4.4. Assuming that the choices of isometric intertwiners $T_e$ , as above, determine a k-graph $\Lambda $ , the family

$$ \begin{align*}\{T_\lambda T^*_\mu: \lambda\in\Lambda^m, \mu\in\Lambda^n, r(\lambda)=r(\mu)=\iota, s(\lambda)=s(\mu)\}\end{align*} $$

is a basis for $Hom(\rho ^n, \rho ^m)$ and each $T_\lambda T^*_\mu $ is a partial isometry.

Theorem 4.5. Consider $\rho _1,\ldots , \rho _k$ finite-dimensional unitary representations of a compact group G and let $\Lambda $ be the k-colored graph with $\Lambda ^0=R\subseteq \hat {G}$ and edges $\Lambda ^{\varepsilon _i}$ determined by the incidence matrices $M_i$ defined above. Assume that the factorization rules determined by the choices of $T_e\in Hom(s(e),r(e)\otimes \rho _i)$ for all edges $e\in \Lambda ^{\varepsilon _i}$ satisfy the associativity condition, so $\Lambda $ becomes a rank k graph. If we consider $P\in C^*(\Lambda )$ ,

$$ \begin{align*}P=\sum_{\lambda\in\iota \Lambda^{(1,\ldots,1)}}S_\lambda S_\lambda^*,\end{align*} $$

where $\iota $ is the trivial representation, then there is a $*$ -isomorphism of the Doplicher–Roberts algebra $\mathcal {O}_{\rho _1,\ldots ,\rho _k}$ onto the corner $PC^*(\Lambda )P$ .

Proof. Since $C^*(\Lambda )$ is generated by linear combinations of $S_\lambda S_\mu ^*$ with $s(\lambda )=s(\mu )$ (see Lemma 3.1 in [Reference Kumjian, Pask, Raeburn and Renault18]), we first define the maps

$$ \begin{align*}\phi_{n,m}:Hom(\rho^n, \rho^m)\to C^*(\Lambda),\quad \phi_{n,m}(T_\lambda T_\mu^*)=S_\lambda S_\mu^*,\end{align*} $$

where $s(\lambda )=s(\mu )$ and $r(\lambda )=r(\mu )=\iota $ . Since $S_\lambda S_\mu ^*=PS_\lambda S_\mu ^*P$ , the maps $\phi _{n,m}$ take values in $PC^*(\Lambda )P$ . We claim that, for any $r\in \mathbb {N}^k$ ,

$$ \begin{align*}\phi_{n+r,m+r}(T_\lambda T_\mu^*\otimes I_r)=\phi_{n,m}(T_\lambda T_\mu^*).\end{align*} $$

This is because

$$ \begin{align*}\mathcal{H}_{s(\lambda)}\otimes \mathcal{H}^r=\bigoplus_{\nu\in s(\lambda)\Lambda^r}T_\nu T_\nu^*(\mathcal{H}_{s(\lambda)}\otimes \mathcal{H}^r),\end{align*} $$

so that

$$ \begin{align*}T_\lambda T_\mu^*\otimes I_r=\sum_{\nu\in s(\lambda)\Lambda^r}(T_\lambda\otimes I_r)(T_\nu T_\nu^*)(T_\mu^*\otimes I_r)=\sum_ {\nu\in s(\lambda)\Lambda^r} T_{\lambda\nu}T_{\mu\nu}^*\end{align*} $$

and

$$ \begin{align*}S_\lambda S_\mu^*=\sum_{\nu\in s(\lambda)\Lambda^r}S_\lambda(S_\nu S_\nu^*)S_\mu^*=\sum_{\nu\in s(\lambda)\Lambda^r}S_{\lambda\nu}S_{\mu\nu}^*.\end{align*} $$

The maps $\phi _{n,m}$ determine a map $\phi : {}^0\mathcal {O}_{\rho _1,\ldots ,\rho _k}\to PC^*(\Lambda )P$ which is linear, $*$ -preserving and multiplicative. Indeed,

$$ \begin{align*}\phi_{n,m}(T_\lambda T_\mu^*)^*=(S_\lambda S_\mu^*)^*=S_\mu S_\lambda^*=\phi_{m,n}(T_\mu T_\lambda^*).\end{align*} $$

Consider now $T_\lambda T_\mu ^*\in Hom(\rho ^n, \rho ^m),\;\; T_\nu T_\omega ^*\in Hom (\rho ^q,\rho ^p)$ with $s(\lambda )=s(\mu ), s(\nu )=s(\omega ), r(\lambda )=r(\mu )=r(\nu )=r(\omega )=\iota $ . Since, for all $n\in \mathbb {N}^k$ ,

$$ \begin{align*}\sum_{\lambda\in\iota\Lambda^n}T_\lambda T_\lambda^*=I_n,\end{align*} $$

we get

$$ \begin{align*}T_\mu^*T_\nu= \begin{cases} T_\beta^*\quad& \text{if } \mu=\nu\beta,\\ T_\alpha \quad& \text{if } \nu=\mu\alpha,\\0 \quad&\text{otherwise,}\end{cases}\end{align*} $$

and hence

$$ \begin{align*}\phi((T_\lambda T_\mu^*)(T_\nu T_\omega^*))=\begin{cases} \phi(T_\lambda T_{\omega\beta}^*)=S_\lambda S_{\omega\beta}^*\quad&\text{if } \mu=\nu\beta,\\ \phi(T_{\lambda\alpha}T_\omega^*)=S_{\lambda\alpha}S_\omega^*\quad&\text{if } \nu=\mu\alpha,\\0 \quad&\text{otherwise.}\end{cases}\end{align*} $$

On the other hand, from Lemma 3.1 in [Reference Kumjian, Pask, Raeburn and Renault18],

$$ \begin{align*}S_\lambda S_\mu^* S_\nu S_{\omega}^*=\begin{cases} S_\lambda S_{\omega\beta}^*\quad&\text{if } \mu=\nu\beta,\\ S_{\lambda\alpha}S_\omega^*\quad&\text{if } \nu=\mu\alpha,\\0 \quad&\text{otherwise,}\end{cases}\end{align*} $$

and hence

$$ \begin{align*}\phi((T_\lambda T_\mu^*)(T_\nu T_\omega^*))=\phi(T_\lambda T_\mu^*)\phi(T_\nu T_\omega^*).\end{align*} $$

Since $PS_\lambda S_\mu ^*P=\phi _{n,m}(T_\lambda T_\mu ^*)$ if $r(\lambda )=r(\mu )=\iota $ and $s(\lambda )=s(\mu )$ , it follows that $\phi $ is surjective. Injectivity follows from the fact that $\phi $ is equivariant for the gauge action.

5 Examples

Example 5.1. Let $G=S_3$ be the symmetric group with $\hat {G}=\{\iota , \epsilon , \sigma \}$ and character table

Here $\iota $ denotes the trivial representation, $\epsilon $ is the sign representation and $\sigma $ is an irreducible $2$ -dimensional representation, for example,

$$ \begin{align*}\sigma((12))=\left[\begin{array}{rr}-1&-1\\0&1\end{array}\right],\quad\sigma((123))=\left[\begin{array}{rr}-1&-1\\1&0\end{array}\right].\end{align*} $$

By choosing $\rho _1=\sigma $ on $\mathcal {H}_1=\mathbb {C}^2$ and $\rho _2=\iota +\sigma $ on $\mathcal {H}_2=\mathbb {C}^3$ , we get a product system $\mathcal {E}\to \mathbb {N}^2$ and an action of $S_3$ on $\mathcal {O}(\mathcal {E})\cong \mathcal O_2\otimes \mathcal O_3$ with fixed point algebra $\mathcal {O}(\mathcal {E})^{S_3}\cong \mathcal {O}_{\rho _1,\rho _2}$ isomorphic to a corner of the $C^*$ -algebra of a rank two graph $\Lambda $ . The set of vertices is $\Lambda ^0=\{\iota ,\epsilon , \sigma \}$ and the edges are given by the incidence matrices

$$ \begin{align*} M_1=\left[\begin{array}{ccc}0&0&1\\0&0&1\\1&1&1\end{array}\right], \quad M_2=\left[\begin{array}{ccc}1&0&1\\0&1&1\\1&1&2\end{array}\right].\end{align*} $$

This is because

$$ \begin{align*}\iota\otimes\rho_1=\sigma,\; \epsilon\otimes \rho_1=\sigma,\; \sigma\otimes \rho_1=\iota+\epsilon+\sigma,\end{align*} $$

$$ \begin{align*}\iota\otimes\rho_2=\iota+\sigma,\; \epsilon\otimes\rho_2=\epsilon+\sigma,\; \sigma\otimes\rho_2=\iota+\epsilon+2\sigma.\end{align*} $$

We label the blue (solid) edges by $e_1,\ldots , e_5$ and the red (dashed) edges by $f_1,\ldots ,f_8$ as in the figure below.

The isometric intertwiners are

$$ \begin{align*}T_{e_1}:\mathcal{H}_\iota\to \mathcal{H}_\sigma\otimes \mathcal{H}_1, \; T_{e_2}:\mathcal{H}_\sigma\to \mathcal{H}_\iota\otimes \mathcal{H}_1, \;T_{e_3}:\mathcal{H}_\epsilon\to \mathcal{H}_\sigma\otimes \mathcal{H}_1,\end{align*} $$

$$ \begin{align*}T_{e_4}:\mathcal{H}_\sigma\to \mathcal{H}_\epsilon\otimes \mathcal{H}_1,\; T_{e_5}:\mathcal{H}_\sigma\to\mathcal{H}_\sigma\otimes\mathcal{H}_1,\end{align*} $$

$$ \begin{align*}T_{f_1}:\mathcal{H}_\iota\to \mathcal{H}_\iota\otimes\mathcal{H}_2,\; T_{f_2}:\mathcal{H}_\epsilon\to \mathcal{H}_\epsilon\otimes\mathcal{H}_2,\; T_{f_3}:\mathcal{H}_\sigma\to \mathcal{H}_\iota\otimes\mathcal{H}_2,\end{align*} $$

$$ \begin{align*}T_{f_4}:\mathcal{H}_\iota\to \mathcal{H}_\sigma\otimes\mathcal{H}_2,\; T_{f_5}:\mathcal{H}_\sigma\to\mathcal{H}_\epsilon\otimes \mathcal{H}_2,\; T_{f_6}:\mathcal{H}_\epsilon\to\mathcal{H}_\sigma\otimes \mathcal{H}_2,\end{align*} $$

$$ \begin{align*} T_{f_7}, T_{f_8}:\mathcal{H}_\sigma\to \mathcal{H}_\sigma\otimes\mathcal{H}_2\end{align*} $$

such that

$$ \begin{align*}T_{e_1}T_{e_1}^*+T_{e_3}T_{e_3}^*+T_{e_5}T_{e_5}^*=I_\sigma\otimes I_1, \; T_{e_2}T_{e_2}^*=I_\iota\otimes I_1,\; T_{e_4}T_{e_4}^*=I_\epsilon\otimes I_1,\end{align*} $$

$$ \begin{align*}T_{f_1}T_{f_1}^*+T_{f_3}T_{f_3}^*=I_\iota\otimes I_2,\; T_{f_2}T_{f_2}^*+T_{f_5}T_{f_5}^*=I_\epsilon\otimes I_2,\end{align*} $$

$$ \begin{align*} T_{f_4}T_{f_4}^*+T_{f_6}T_{f_6}^*+T_{f_7}T_{f_7}^*+T_{f_8}T_{f_8}^*=I_\sigma\otimes I_2.\end{align*} $$

Here $I_\pi $ is the identity of $\mathcal {H}_\pi $ for $\pi \in \hat {G}$ and $I_i$ is the identity of $\mathcal {H}_i$ for $ i=1,2$ . Since

$$ \begin{align*}M_1M_2=\left[\begin{array}{ccc}1&1&2\\1&1&2\\2&2&4\end{array}\right]\end{align*} $$

and

$$ \begin{align*}T_{e_2}T_{f_4}, T_{f_3}T_{e_1}\in Hom(\iota, \iota\otimes\rho_1\otimes\rho_2),\end{align*} $$

$$ \begin{align*}T_{e_2}T_{f_6}, T_{f_3}T_{e_3}\in Hom(\epsilon, \iota\otimes\rho_1\otimes\rho_2),\end{align*} $$

$$ \begin{align*}T_{e_2}T_{f_7}, T_{e_2}T_{f_8}, T_{f_1}T_{e_2}, T_{f_3}T_{e_5}\in Hom(\sigma, \iota\otimes\rho_1\otimes\rho_2),\end{align*} $$

$$ \begin{align*}T_{e_4}T_{f_4}, T_{f_5}T_{e_1}\in Hom(\iota, \epsilon\otimes\rho_1\otimes\rho_2),\end{align*} $$

$$ \begin{align*}T_{e_4}T_{f_6}, T_{f_5}T_{e_3}\in Hom(\epsilon, \epsilon\otimes\rho_1\otimes\rho_2),\end{align*} $$

$$ \begin{align*}T_{e_4}T_{f_7}, T_{e_4}T_{f_8}, T_{f_2}T_{e_4}, T_{f_5}T_{e_5}\in Hom(\sigma, \epsilon\otimes\rho_1\otimes\rho_2),\end{align*} $$

$$ \begin{align*}T_{e_1}T_{f_1}, T_{e_5}T_{f_4}, T_{f_7}T_{e_1}, T_{f_8}T_{e_1}\in Hom(\iota, \sigma\otimes\rho_1\otimes\rho_2),\end{align*} $$

$$ \begin{align*}T_{e_3}T_{f_2}, T_{e_5}T_{f_6}, T_{f_7}T_{e_3}, T_{f_8}T_{e_3}\in Hom(\epsilon,\sigma\otimes\rho_1\otimes\rho_2),\end{align*} $$

$$ \begin{align*}T_{e_5}T_{f_7}, T_{e_5}T_{f_8}, T_{e_3}T_{f_5}, T_{e_1}T_{f_3}, T_{f_6}T_{e_4}, T_{f_4}T_{e_2}, T_{f_7}T_{e_5}, T_{f_8}T_{e_5}\in Hom(\sigma, \sigma\otimes\rho_1\otimes\rho_2),\end{align*} $$

a possible choice of commuting squares is

$$ \begin{align*}e_2f_4=f_3e_1,\; e_2f_6=f_3e_3,\; e_2f_7=f_1e_2,\; e_2f_8=f_3e_5,\; e_4f_4=f_5e_1,\; e_4f_6=f_5e_3,\end{align*} $$

$$ \begin{align*}e_4f_7= f_2e_4,\; e_4f_8=f_5e_5,\; e_1f_1=f_7e_1,\; e_5f_4=f_8e_1,\; e_3f_2=f_7e_3,\; e_5f_6=f_8e_3,\end{align*} $$

$$ \begin{align*}e_5f_7=f_6e_4,\; e_5f_8=f_4e_2,\; e_3f_5=f_7e_5,\; e_1f_3=f_8e_5.\end{align*} $$

This data is enough to determine a rank two graph $\Lambda $ associated to $\rho _1, \rho _2$ . But this is not the only choice, since, for example, we could have taken

$$ \begin{align*}e_2f_4=f_3e_1,\; e_2f_6=f_3e_3,\; e_2f_8=f_1e_2,\; e_2f_7=f_3e_5,\; e_4f_4=f_5e_1,\; e_4f_6=f_5e_3,\end{align*} $$

$$ \begin{align*}e_4f_8= f_2e_4,\; e_4f_7=f_5e_5,\; e_1f_1=f_7e_1,\; e_5f_4=f_8e_1,\; e_3f_2=f_8e_3,\; e_5f_6=f_7e_3,\end{align*} $$

$$ \begin{align*}e_5f_7=f_6e_4,\; e_5f_8=f_4e_2,\; e_3f_5=f_7e_5,\; e_1f_3=f_8e_5,\end{align*} $$

which determines a different $2$ -graph.

A direct analysis using the definitions shows that, in each case, the $2$ -graph $\Lambda $ is cofinal, satisfies the aperiodicity condition and every vertex connects to a loop with an entrance. It follows that $C^*(\Lambda )$ is simple and purely infinite and the Doplicher–Roberts algebra $\mathcal {O}_{\rho _1,\rho _2}$ is Morita equivalent with $C^*(\Lambda )$ .

The K-theory of $C^*(\Lambda )$ can be computed using Proposition 3.16 in [Reference Evans11] and it does not depend on the choice of factorization rules. We have

$$ \begin{align*}K_0(C^*(\Lambda))\cong\text{coker}[I-M_1^t\;\; I-M_2^t]\oplus\ker\left[\begin{array}{c}M_2^t-I\\I-M_1^t\end{array}\right]\cong \mathbb Z/2\mathbb Z,\end{align*} $$

$$ \begin{align*}K_1(C^*(\Lambda))\cong\ker[I-M_1^t\;\;I-M_2^t]/\text{im}\left[\begin{array}{c}M_2^t-I\\I-M_1^t\end{array}\right]\cong 0.\end{align*} $$

In particular, $\mathcal {O}_{\rho _1,\rho _2}\cong \mathcal {O}_3$ .

On the other hand, since $\rho _1, \rho _2$ are faithful, both Doplicher–Roberts algebras $\mathcal {O}_{\rho _1}, \mathcal {O}_{\rho _2}$ are simple and purely infinite with

$$ \begin{align*}K_0(\mathcal{O}_{\rho_1})\cong \mathbb{Z}/2\mathbb{Z},\; K_1(\mathcal{O}_{\rho_1})\cong 0,\; K_0(\mathcal{O}_{\rho_2})\cong \mathbb{Z},\; K_1(\mathcal{O}_{\rho_2})\cong \mathbb{Z},\end{align*} $$

so $\mathcal {O}_{\rho _1,\rho _2}\ncong \mathcal {O}_{\rho _1}\otimes \mathcal {O}_{\rho _2}$ .

Example 5.2. With $G=S_3$ and $\rho _1=2\iota , \rho _2=\iota +\epsilon $ , then $R=\{\iota , \epsilon \}$ , so $\Lambda $ has two vertices and incidence matrices

$$ \begin{align*}M_1=\left[\begin{array}{cc}2&0\\0&2\end{array}\right],\quad M_2=\left[\begin{array}{cc}1&1\\1&1\end{array}\right],\end{align*} $$

which give

Again, a corresponding choice of isometric intertwiners determines some factorization rules, for example,

$$ \begin{align*}e_1f_1=f_1e_2,\; e_2f_1=f_1e_1,\; e_1f_3=f_3e_3,\; e_2f_3=f_3e_4,\end{align*} $$

$$ \begin{align*}e_3f_2=f_2e_1,\; e_4f_2=f_2e_2,\; e_3f_4=f_4e_4,\; e_4f_4=f_4e_3.\end{align*} $$

Even though $\rho _1, \rho _2$ are not faithful, the obtained $2$ -graph is cofinal, satisfies the aperiodicity condition and every vertex connects to a loop with an entrance, so $\mathcal {O}_{\rho _1,\rho _2}$ is simple and purely infinite with trivial K-theory. In particular, $\mathcal {O}_{\rho _1,\rho _2}\cong \mathcal {O}_2$ .

Note that, since $\rho _1, \rho _2$ have kernel $N=\langle (123)\rangle \cong \mathbb {Z}/3\mathbb {Z}$ , we could replace G by $G/N\cong \mathbb {Z}/2\mathbb {Z}$ and consider $\rho _1,\rho _2$ as representations of $\mathbb {Z}/2\mathbb {Z}$ .

Example 5.3. Consider $G=\mathbb {Z}/2\mathbb {Z}=\{0,1\}$ with $\hat {G}=\{\iota ,\chi \}$ and character table

Choose the $2$ -dimensional representations

$$ \begin{align*}\rho_1=\iota+\chi,\; \rho_2=2\iota,\; \rho_3=2\chi,\end{align*} $$

which determine a product system $\mathcal {E}$ such that $\mathcal {O}(\mathcal {E})\cong \mathcal {O}_2\otimes \mathcal {O}_2\otimes \mathcal {O}_2$ and a Doplicher–Roberts algebra $\mathcal {O}_{\rho _1,\rho _2,\rho _3}\cong \mathcal {O}(\mathcal {E})^{\mathbb {Z}/2\mathbb {Z}}$ .

An easy computation shows that the incidence matrices of the blue (solid), red (dashed) and green (dotted) graphs are

$$ \begin{align*}M_1=\left[\begin{array}{cc}1&1\\1&1\end{array}\right],\quad M_2=\left[\begin{array}{cc}2&0\\0&2\end{array}\right],\quad M_3=\left[\begin{array}{cc}0&2\\2&0\end{array}\right].\end{align*} $$

With labels as in the figure, we choose the following factorization rules.

$$ \begin{align*}e_1f_1=f_2e_1,\; e_1f_2=f_1e_1,\; e_2f_1=f_4e_2,\; e_2f_2=f_3e_2,\end{align*} $$

$$ \begin{align*}e_3f_3=f_2e_3,\; e_3f_4=f_1e_3,\; e_4f_4=f_3e_4,\; e_4f_3=f_4e_4,\end{align*} $$

$$ \begin{align*}f_1g_1=g_2f_3,\; f_1g_2=g_1f_3,\; f_2g_1=g_2f_4,\; f_2g_2=g_1f_4,\end{align*} $$

$$ \begin{align*}f_3g_3=g_4f_1,\; f_3g_4=g_3f_1,\; f_4g_3=g_4f_2,\; f_4g_4=g_3f_2,\end{align*} $$

$$ \begin{align*}e_1g_1=g_2e_4,\; e_1g_2=g_1e_4,\; e_2g_1=g_3e_3,\; e_2g_2=g_4e_3,\end{align*} $$

$$ \begin{align*}e_3g_3=g_1e_2,\; e_3g_4=g_2e_2,\; e_4g_3=g_4e_1,\; e_4g_4=g_3e_1.\end{align*} $$

A tedious verification shows that all the following paths are well defined.

$$ \begin{align*}e_1f_1g_1,\; e_1f_1g_2,\; e_1f_2g_1, \; e_1f_2g_2,\; e_2f_1g_1,\; e_2f_1g_2,\; e_2f_2g_1,\; e_2f_2g_2,\end{align*} $$

$$ \begin{align*}e_3f_3g_3,\; e_3f_3g_4,\; e_3f_4g_3,\; e_3f_4g_4,\; e_4f_3g_3,\; e_4f_3g_4,\; e_4f_4g_3,\; e_4f_4g_4,\end{align*} $$

so the associativity property is satisfied and we get a rank three graph $\Lambda $ with two vertices. It is not difficult to check that $\Lambda $ is cofinal, satisfies the aperiodicity condition and every vertex connects to a loop with an entrance, so $C^*(\Lambda )$ is simple and purely infinite.

Since $\partial _1=[I-M_1^t\; I-M_2^t\; I-M_3^t]:\mathbb {Z}^6\to \mathbb {Z}^2$ is surjective, using Corollary 3.18 in [Reference Evans11], we obtain

$$ \begin{align*}K_0(C^*(\Lambda))\cong \ker\partial_2/\text{im}\; \partial_3\cong 0,\;K_1(C^*(\Lambda))\cong \ker\partial_1/\text{im}\; \partial_2\oplus \ker\partial_3\cong 0,\end{align*} $$

where

$$ \begin{align*}\partial_2=\left[\begin{array}{ccc} M_2^t-I&M_3^t-I&0\\I-M_1^t&0&M_3^t-I\\0&I-M_1^t&I-M_2^t\end{array}\right],\quad\partial_3=\left[\begin{array}{c}I-M_3^t\\M_2^t-I\\I-M_1^t\end{array}\right],\end{align*} $$

and, in particular, $\mathcal {O}_{\rho _1,\rho _2,\rho _3}\cong \mathcal {O}_2$ .

Example 5.4. Let $G=\mathbb {T}$ . We have $\hat {G}=\{\chi _k:k\in \mathbb {Z}\}$ , where $\chi _k(z)=z^k$ and $\chi _k\otimes \chi _\ell =\chi _{k+\ell }$ . The faithful representations

$$ \begin{align*}\rho_1=\chi_{-1}+\chi_0,\; \rho_2=\chi_0+\chi_1\end{align*} $$

of $\mathbb {T}$ determine a product system $\mathcal {E}$ with $\mathcal {O}(\mathcal {E})\cong \mathcal {O}_2\otimes \mathcal {O}_2$ and a Doplicher–Roberts algebra $\mathcal {O}_{\rho _1,\rho _2}\cong \mathcal {O}(\mathcal {E})^{\mathbb {T}}$ isomorphic to a corner in the $C^*$ -algebra of a rank $2$ graph $\Lambda $ with $\Lambda ^0=\hat {G}$ and infinite incidence matrices, where

$$ \begin{align*}M_1(\chi_k,\chi_\ell)=\begin{cases}1 \quad&\text{if } \ell=k \;\text{or}\; \ell=k-1,\\0\quad& \text{otherwise,}\end{cases}\end{align*} $$

$$ \begin{align*}M_2(\chi_k,\chi_\ell)=\begin{cases} 1 \quad&\text{if } \ell=k \;\text{or}\; \ell=k+1,\\0\quad& \text{otherwise.}\end{cases}\end{align*} $$

The skeleton of $\Lambda $ looks like

and this $2$ -graph is cofinal, satisfies the aperiodicity condition and every vertex connects to a loop with an entrance, so $C^*(\Lambda )$ is simple and purely infinite.

Example 5.5. Let $G=SU(2)$ . It is known (see page 84 in [Reference Bröcker and tom Dieck2]) that the elements in $\hat {G}$ are labeled by $V_n$ for $n\ge 0$ , where $V_0=\iota $ is the trivial representation on $\mathbb {C}$ , $V_1$ is the standard representation of $SU(2)$ on $\mathbb {C}^2$ , and, for $n\ge 2$ , $V_n=S^nV_1$ , the $n\, $ th symmetric power. In fact, $\dim V_n=n+1$ and $V_n$ can be taken as the representation of $SU(2)$ on the space of homogeneous polynomials p of degree n in variables $z_1,z_2$ , where, for $ g=\big [\begin {smallmatrix}a&b\\c&d\end {smallmatrix}\big ]\in SU(2)$ ,

$$ \begin{align*}(g\cdot p)(z)=p(az_1+cz_2, bz_1+dz_2).\end{align*} $$

The irreducible representations $V_n$ satisfy the Clebsch–Gordan formula

$$ \begin{align*}V_k\otimes V_\ell=\bigoplus_{j=0}^qV_{k+\ell-2j},\; q=\min\{k,\ell\}.\end{align*} $$

If we choose $\rho _1=V_1, \rho _2=V_2$ , then we get a product system $\mathcal {E}$ with $\mathcal {O}(\mathcal {E})\cong \mathcal {O}_2\otimes \mathcal {O}_3$ and a Doplicher–Roberts algebra $\mathcal {O}_{\rho _1,\rho _2}\cong \mathcal {O}(\mathcal {E})^{SU(2)}$ isomorphic to a corner in the $C^*$ -algebra of a rank two graph with $\Lambda ^0=\hat {G}$ and edges given by the matrices

$$ \begin{align*}M_1(V_k,V_\ell)=\begin{cases}1\quad&\text{if } k=0\;\text{and}\; \ell=1,\\ 1\quad& \text{if } k\ge 1\;\text{and}\; \ell \in\{k-1,k+1\},\\0\quad&\text{otherwise,}\end{cases}\end{align*} $$

$$ \begin{align*}M_2(V_k, V_\ell)=\begin{cases} 1 \quad&\text{if } k=0\;\text{and}\; \ell=2,\\1\quad&\text{if } k=1\;\text{and}\; \ell\in \{1,3\},\\ 1\quad&\text{if } k\ge 2\;\text{and}\; \ell\in\{k-2,k,k+2\},\\0\quad&\text{otherwise.}\end{cases}\end{align*} $$

The skeleton looks like

and this $2$ -graph is cofinal, satisfies the aperiodicity condition and every vertex connects to a loop with an entrance; in particular, $\mathcal {O}_{\rho _1,\rho _2}$ is simple and purely infinite.