$\boldsymbol {C}^{*}$
 -ALGEBRAS FROM 
$\boldsymbol {K}$
 GROUP REPRESENTATIONS

Abstract We introduce certain 
$C^*$
 -algebras and k-graphs associated to k finite-dimensional unitary representations 
$\rho _1,\ldots ,\rho _k$
 of a compact group G. We define a higher rank Doplicher-Roberts algebra 
$\mathcal {O}_{\rho _1,\ldots ,\rho _k}$
 , constructed from intertwiners of tensor powers of these representations. Under certain conditions, we show that this 
$C^*$
 -algebra is isomorphic to a corner in the 
$C^*$
 -algebra of a row-finite rank k graph 
$\Lambda $
 with no sources. For G finite and 
$\rho _i$
 faithful of dimension at least two, this graph is irreducible, it has vertices 
$\hat {G}$
 and the edges are determined by k commuting matrices obtained from the character table of the group. We illustrate this with some examples when 
$\mathcal {O}_{\rho _1,\ldots ,\rho _k}$
 is simple and purely infinite, and with some K-theory computations.


Introduction
The study of graph C * -algebras was motivated, among other reasons, by the Doplicher-Roberts algebra O ρ associated to a group representation ρ (see [19,22]). It is natural to imagine that a rank k graph is related to a fixed set of k representations ρ 1 , . . . , ρ k satisfying certain properties.
There are various extra conditions on a product system E → 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 O A (E) or the Cuntz-Nica-Pimsner algebra NO A (E) satisfying certain properties (see [1,4,10,13,26], among others). We mention that O A (E) (or NO A (E)) comes with a covariant representation j E : E → O A (E) and is universal in the following sense: O A (E) is generated by j E (E) and, for any covariant representation ψ : E → B, there is a homomorphism ψ * : O A (E) → B such that ψ * • j E = ψ. Under certain conditions, O A (E) satisfies a gauge invariant uniqueness theorem. EXAMPLE 2.3. For a product system E → P with fibers E p that are nonzero finite-dimensional Hilbert spaces, and, in particular, A = E e = C, let us fix an orthonormal basis B p in E p . Then a Toeplitz representation ψ : E → B gives rise to a family of isometries {ψ(ξ) : ξ ∈ B p } p∈P with mutually orthogonal range projections. In this case, T (E) = T C (E) is generated by a collection of Cuntz-Toeplitz algebras which interact according to the multiplication maps M p,q in E.
A representation ψ : E → B is Cuntz-Pimsner covariant if ξ∈B p ψ(ξ)ψ(ξ) * = ψ (1) for all p ∈ P. The Cuntz-Pimsner algebra O(E) = O C (E) is generated by a collection of Cuntz algebras, so it could be thought of as a multidimensional Cuntz algebra. Fowler proved in [12] that if the function p → dim E p is injective, then the algebra O(E) is simple and purely infinite. For other examples of multidimensional Cuntz algebras, see [3]. EXAMPLE 2.4. A row-finite k-graph with no sources Λ (see [18]) determines a product system E → N k with E 0 = A = C 0 (Λ 0 ) and E n = C c (Λ n ) for n 0 such that we have a T k -equivariant isomorphism O A (E) C * (Λ). Recall that, for product systems indexed by N k , the universal property induces a gauge action on O A (E) defined by γ z (j E (ξ)) = z n j E (ξ) for z ∈ T k and ξ ∈ E n .
The following two definitions and two results are taken from [7]; see also [15,17]. DEFINITION 2.5. An action β of a locally compact group G on a product system E → P over A is a family (β p ) p∈P such that β p is an action of G on each fiber E p compatible with the action α = β e on A, and, furthermore, the actions (β p ) p∈P are compatible with the multiplication maps M p,q in the sense that β pq g (M p,q (x ⊗ y)) = M p,q (β p g (x) ⊗ β q g (y)) for all g ∈ G, x ∈ E p and y ∈ E q . DEFINITION 2.6. If β is an action of G on the product system E → P, we define the crossed product E β G as the product system indexed by P with fibers E p β p G, which are C * -correspondences over A α G. For ζ ∈ C c (G, E p ) and η ∈ C c (G, E q ), the product The set E β G = p∈P E p β p G with the above multiplication satisfies all the properties of a product system of C * -correspondences over A α G. PROPOSITION 2.8. Suppose that a locally compact group G acts on a row-finite and faithful product system E indexed by P = (N k , +) via automorphisms β p g . Then G acts on the Cuntz-Pimsner algebra O A (E) via automorphisms denoted by γ g . Moreover, if G is amenable, then E β G is row-finite and faithful, and 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 ρ i of G on Hilbert spaces H i for i = 1, . . . , k, we construct the product system E = E(ρ 1 , . . . , ρ k ) indexed by the commutative monoid (N k , +), with fibers E n = H n = H ⊗n 1 1 ⊗ · · · ⊗ H ⊗n k k for n = (n 1 , . . . , n k ) ∈ N k ; in particular, A = E 0 = C. The multiplication maps M n,m : E n × E m → E n+m in E are defined by using the standard isomorphisms ρ i ⊗ ρ j ρ j ⊗ ρ i for all i < j. The associativity in E follows from the fact that as maps from E n × E m × E p to E n+m+p . Then E = E(ρ 1 , . . . , ρ k ) is called the product system of the representations ρ 1 , . . . , ρ k . REMARK 2.10. Similarly, a semigroup P of unitary representations of a group G determines a product system E → P. PROPOSITION 2.11. With notation as in Definition 2.9, assume that Then the Cuntz-Pimsner algebra O(E) associated to the product system E → N k described above is isomorphic with the C * -algebra of a rank k graph Γ with a single vertex and with |Γ ε i | = d i . This isomorphism is equivariant for the gauge action. Moreover, where O n is the Cuntz algebra. PROOF. Indeed, by choosing a basis in each H i , we get the edges Γ ε i in a k-colored graph Γ with a single vertex. The isomorphisms ρ i ⊗ ρ j ρ j ⊗ ρ i determine the factorization rules of the form e f = f e for e ∈ Γ ε i and f ∈ Γ ε j , which obviously satisfy the associativity condition. In particular, the corresponding isometries in C * (Γ) commute and determine, by the universal property, a surjective homomorphism ϕ onto O(E), preserving the gauge action. Using the gauge invariant uniqueness theorem for k-graph algebras, the map ϕ is an isomorphism. In particular, is always simple and purely infinite since it is a tensor product of simple and purely infinite C * -algebras. If d i = 1 for some i, then the isomorphism in Proposition 2.11 still holds, but C * (Γ) O(E) contains a copy of C(T), so it is not simple. Of course, if d i = 1 for all i, then O(E) C(T k ). For more on single vertex rank k graphs, see [5,6]. PROPOSITION 2.13. The compact group G acts on each fiber E n of the product system E via the representation ρ n = ρ ⊗n 1 1 ⊗ · · · ⊗ ρ ⊗n k k . This action is compatible with the multiplication maps and commutes with the gauge action of T k . The crossed product E G becomes a row-finite and faithful product system indexed by N k over the group C * -algebra C * (G). Moreover, PROOF. Indeed, for g ∈ G and ξ ∈ E n = H n , we define g · ξ = ρ n (g)(ξ), and since ρ i ⊗ ρ j ρ j ⊗ ρ i , we have g · (ξ ⊗ η) = g · ξ ⊗ g · η for ξ ∈ E n , η ∈ E m . Clearly, g · γ z (ξ) = g · (z n ξ) = z n (g · ξ) = γ z (g · ξ), so the action of G commutes with the gauge action. Using Proposition 2.7, E G becomes a product system indexed by N k over C * (G) C G with fibers E n G. The isomorphism O(E) G O C * (G) (E G) follows from Proposition 2.8. COROLLARY 2.14. Since the action of G commutes with the gauge action, the group G acts on the core algebra F = O(E) T k .
where Λ G is a Zappa-Szép product and Q(Λ G) is its boundary quotient C * -algebra (see Example 3.10(4) in [21] and Theorem 3.3 in [20]). I thank the referee for bringing this relationship to my attention.

The Doplicher-Roberts algebra
The Doplicher-Roberts algebras O ρ , denoted by O G in [8], were introduced to construct a new duality theory for compact Lie groups G that strengthens the Tannaka-Krein duality. Here ρ is the n-dimensional representation of G defined by the inclusion G ⊆ U(n) in some unitary group U(n). Let T G denote the representation category whose objects are tensor powers ρ p = ρ ⊗p for p ≥ 0, and whose arrows are the intertwiners Hom(ρ p , ρ q ). The group G acts via ρ on the Cuntz algebra O n and O G = O ρ is identified in [8] with the fixed point algebra O G n . If σ denotes the restriction to O ρ of the canonical endomorphism of O n , then T G can be reconstructed from the pair (O ρ , σ). Subsequently, Doplicher-Roberts algebras were associated to any object ρ in a strict tensor C * -category (see [9]).
Then the linear span of A 0 becomes a * -algebra A with appropriate multiplication and involution. This algebra has a natural Z k -grading coming from a gauge action of T k . Moreover, the Cuntz-Pimsner algebra O(E) of the product system E = E(ρ 1 , . . . , ρ k ) is equivariantly isomorphic to the C * -closure of A in the unique C * -norm for which the gauge action is isometric.
PROOF. Recall that the Cuntz algebra O n contains a canonical Hilbert space H of dimension n and it can be constructed as the closure of the linear span of where H p = H ⊗p and I : H → H is the identity map. This linear span becomes a * -algebra with a multiplication given by composition and an involution (see [8] and Proposition 2.5 in [16]). Similarly, for all r ∈ N k , we consider embeddings L(H n , H m ) ⊆ L(H n+r , H m+r ) given by T → T ⊗ I r , where I r : H r → H r is the identity map, and we endow A with a multiplication given by composition and an involution. More precisely, if S ∈ L(H n , H m ) and T ∈ L(H q , H p ), then the product ST is where we write p ∨ n for the coordinatewise maximum. This multiplication is well defined in A and is associative.
There is a natural Z k -grading on A given by the gauge action γ of T k , where, for z = (z 1 , . . . , z k ) ∈ T k and T ∈ L(H n , H m ), we define Adapting the argument in Theorem 4.2 in [9] for Z k -graded C * -algebras, the C * -closure of A in the unique C * -norm for which γ z is isometric is well defined. The map . . , k preserves the gauge action and it can be extended is the fixed point algebra under the gauge action, which is a UHF-algebra.
To define the Doplicher-Roberts algebra O ρ 1 ,...,ρ k , we again identify Hom(ρ n , ρ m ) with a subset of Hom(ρ n+r , ρ m+r ) for each r ∈ N k , via T → T ⊗ I r . After this identification, it follows that the linear span 0 O ρ 1 ,...,ρ k of m,n∈N k Hom(ρ n , ρ m ) ⊆ A 0 has a natural multiplication and involution inherited from A. Indeed, a computation shows that if S ∈ Hom(ρ n , ρ m ) and T ∈ Hom(ρ q , ρ p ), then S * ∈ Hom(ρ m , ρ n ) and ..,ρ k is closed under these operations. Since the action of G commutes with the gauge action, there is a natural Z k -grading of 0 O ρ 1 ,...,ρ k given by the gauge action γ of T k on A.
It follows that the closure O ρ 1 ,...,ρ k of 0 O ρ 1 ,...,ρ k in O(E) is well defined, obtaining the Doplicher-Roberts algebra associated to the representations ρ 1 , . . . , ρ k . This C * -algebra also has a Z k -grading and a gauge action of T k . By construction, O ρ 1 ,...,ρ k ⊆ O(E). REMARK 3.2. For a compact Lie group G, our Doplicher-Roberts algebra O ρ 1 ,...,ρ k is Morita equivalent with the higher rank Doplicher-Roberts algebra D defined in [1]. It is also the section C * -algebra of a Fell bundle over Z k .
PROOF. We know from Lemma 3.1 that O(E) is isomorphic to the C * -algebra generated by the linear span of A 0 = m,n∈N k L(H n , H m ). The group G acts on and the fixed point set is Hom(ρ n , ρ m ). Indeed, we have g · T = T if and only if Tρ n (g) = ρ m (g)T. This action is compatible with the embeddings and the operations, so it extends to the * -algebra A and the fixed point algebra is the linear span of REMARK 3.4. By left tensoring with I r for r ∈ N k , we obtain some canonical unital endomorphisms σ r of O ρ 1 ,...,ρ k .
In the next section, we show that, in many cases, O ρ 1 ,...,ρ k is isomorphic to a corner of C * (Λ) for a rank k graph Λ, so, in some cases, we can compute its K-theory. It would be nice to express the K-theory of O ρ 1 ,...,ρ k in terms of the maps π → π ⊗ ρ i defined on the representation ring R(G).

The rank k graphs
For convenience, we first collect some facts about higher rank graphs, introduced in [18]. A rank k graph or k-graph (Λ, d) consists of a countable small category Λ with range and source maps r and s together with a functor d : Λ → N k called the degree map, satisfying the factorization property: for every λ ∈ Λ and all m, n ∈ N k with d(λ) = m + n, there are unique elements μ, ν ∈ Λ such that λ = μν and d(μ) = m, d(ν) = n. For n ∈ N k , we write Λ n := d −1 (n) and call it the set of paths of degree n. For ε i = (0, . . . , 1, . . . , 0) with 1 in position i, the elements in Λ ε i are called edges and the elements in Λ 0 are called vertices.
A k-graph Λ can be constructed from Λ 0 and from its k-colored skeleton Λ ε 1 ∪ · · · ∪ Λ ε k using a complete and associative collection of commuting squares or factorization rules (see [25]).
The k-graph Λ is row-finite if, for all n ∈ N k and all v ∈ Λ 0 , the set vΛ n := {λ ∈ Λ n : r(λ) = v} is finite. It has no sources if vΛ n ∅ for all v ∈ Λ 0 and n ∈ N k . A k-graph Λ is said to be irreducible (or strongly connected) if, for every u, v ∈ Λ 0 , there is λ ∈ Λ such that u = r(λ) and v = s(λ).
Recall that C * (Λ) is the universal C * -algebra generated by a family {S λ : λ ∈ Λ} of partial isometries satisfying: is a family of mutually orthogonal projections; • S λμ = S λ S μ for all λ, μ ∈ Λ such that s(λ) = r(μ); • S * λ S λ = S s(λ) for all λ ∈ Λ; and • for all v ∈ Λ 0 and n ∈ N k , A k-graph Λ is said to satisfy the aperiodicity condition if, for every vertex v ∈ Λ 0 , there is an infinite path x ∈ vΛ ∞ such that σ m x σ n x for all m n in N k , where σ m : Λ ∞ → Λ ∞ are the shift maps. We say that Λ is cofinal if, for every x ∈ Λ ∞ and v ∈ Λ 0 , there is λ ∈ Λ and n ∈ N k such that s(λ) = x(n) and r(λ) = v.
Assume that Λ is row-finite with no sources and that it satisfies the aperiodicity condition. Then C * (Λ) is simple if and only if Λ is cofinal (see Proposition 4.8 in [18] and Theorem 3.4 in [23]).
We say that a path μ ∈ Λ is a loop with an entrance if s(μ) = r(μ), and there exists α ∈ s(μ)Λ such that d(μ) ≥ d(α) and there is no β ∈ Λ with μ = αβ. We say that every vertex connects to a loop with an entrance if, for every v ∈ Λ 0 , there is a loop with an entrance μ ∈ Λ, and a path λ ∈ Λ with r(λ) = v and s(λ) = r(μ) = s(μ). If Λ satisfies the aperiodicity condition and every vertex connects to a loop with an entrance, then C * (Λ) is purely infinite (see Proposition 4.9 in [18] and Proposition 8.8 in [24]).
Given finite-dimensional unitary representations ρ i of a compact group G on Hilbert spaces H i for i = 1, . . . , k, we want to construct a rank k graph Λ = Λ(ρ 1 , . . . , ρ k ). Let R be the set of equivalence classes of irreducible summands π : G → U(H π ) which appear in the tensor powers ρ n = ρ ⊗n 1 1 ⊗ · · · ⊗ ρ ⊗n k k for n ∈ N k , as in [22]. Take Λ 0 = R and, for each i = 1, . . . , k, consider the set of edges Λ ε i which are uniquely determined by the matrices M i with entries where v, w ∈ R. The matrices M i commute since ρ i ⊗ ρ j ρ j ⊗ ρ i and therefore for all i < j. This allows us to fix some bijections for all 1 ≤ i < j ≤ k, which determine the commuting squares of Λ. As usual, For k ≥ 3, we also need to verify that λ ij can be chosen to satisfy the associativity condition, that is,  PROOF. Indeed, the sets Λ ε i are uniquely determined and the choice of bijections λ ij satisfying the associativity condition is enough to determine Λ. Since the entries of the matrices M i are finite and there are no zero rows, the graph is locally finite with no sources. To prove that Λ is cofinal, fix a vertex v ∈ Λ 0 and an infinite path x ∈ Λ ∞ . Arguing as in Lemma 7.2 in [19], any w ∈ Λ 0 , in particular, w = x(n) for a fixed n, can be joined by a path to v, so there is λ ∈ Λ with s(λ) = x(n) and r(λ) = v. See also Lemma 3.1 in [22].
Here ρ * i denotes the dual representation defined by ρ * i (g) = ρ i (g −1 ) t and equal, in our case, to the conjugate representationρ 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 λ ij are, in general, not unique, the rank k graph Λ is not unique, as illustrated in some examples. It is an open question how the C * -algebra C * (Λ) depends, in general, on the factorization rules.
To relate the Doplicher-Roberts algebra O ρ 1 ,...,ρ k to a rank k graph Λ, we mimic the construction in [22]. For each edge e ∈ Λ ε i , choose an isometric intertwiner T e : H s(e) → H r(e) ⊗ H i in such a way that for all π ∈ Λ 0 , that is, the edges in Λ ε i ending at π give a specific decomposition of H π ⊗ H i into irreducibles. When dim Hom(s(e), r(e) ⊗ ρ i ) ≥ 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 Λ and whether or not they satisfy the associativity condition. Given e ∈ Λ ε i and f ∈ Λ ε j with r( f ) = s(e), we know how to multiply T e ∈ Hom(s(e), r(e) ⊗ ρ i ) with T f ∈ Hom(s( f ), r( f ) ⊗ ρ j ) in the algebra O ρ 1 ,...,ρ k , by viewing Hom(s(e), r(e) ⊗ ρ i ) as a subspace of Hom(ρ n , ρ m ) for some m, n, and similarly for Hom(s( f ), T e ∈ Hom(s(e ), r(e ) ⊗ ρ i ). This is possible since and ρ i ⊗ ρ j ρ j ⊗ ρ i . In this case, we declare that e f = f e . Repeating this process, we obtain bijections λ ij : Assuming that the associativity conditions are satisfied, we obtain a k-graph Λ.
We write T e f = T e T f = T f T e = T f e . A finite path λ ∈ Λ n is a concatenation of edges and determines by composition a unique intertwiner Moreover, the paths λ ∈ Λ n with r(λ) = ι, the trivial representation, provide an explicit decomposition of H n = H ⊗n 1 1 ⊗ · · · ⊗ H ⊗n k k into irreducibles, and hence

PROPOSITION 4.4. Assuming that the choices of isometric intertwiners T e , as above, determine a k-graph Λ, the family
is a basis for Hom(ρ n , ρ m ) and each T λ T * μ is a partial isometry.
PROOF. Each pair of paths λ, μ with d(λ) = m, d(μ) = n and r(λ) = r(μ) = ι determines a pair of irreducible summands T λ (H s(λ) ), T μ (H s(μ) ) of H m and H n , respectively. By Schur's lemma, the space of intertwiners of these representations is trivial unless s(λ) = s(μ), in which case it is the one-dimensional space spanned by T λ T * μ . It follows that any element of Hom(ρ n , ρ m ) can be uniquely represented as a linear combination of elements T λ T * μ , where s(λ) = s(μ). Since T μ is isometric, T * μ is a partial isometry with range H s(μ) and hence T λ T * μ is also a partial isometry whenever s(λ) = s(μ). THEOREM 4.5. Consider ρ 1 , . . . , ρ k finite-dimensional unitary representations of a compact group G and let Λ be the k-colored graph with Λ 0 = R ⊆Ĝ and edges Λ ε i determined by the incidence matrices M i defined above. Assume that the factorization rules determined by the choices of T e ∈ Hom(s(e), r(e) ⊗ ρ i ) for all edges e ∈ Λ ε i satisfy the associativity condition, so Λ becomes a rank k graph. If we consider P ∈ C * (Λ), where ι is the trivial representation, then there is a * -isomorphism of the Doplicher-Roberts algebra O ρ 1 ,...,ρ k onto the corner PC * (Λ)P.
PROOF. This follows from the fact that C * (Λ) is simple and purely infinite and because PC * (Λ)P is a full corner. REMARK 4.7. There is a groupoid G Λ associated to a row-finite rank k graph Λ with no sources (see [18]). By taking the pointed groupoid G Λ (ι), the reduction to the set of infinite paths with range ι, under the same conditions as in Theorem 4.5, we get an isomorphism of the Doplicher-Roberts algebra O ρ 1 ,...,ρ k onto C * (G Λ (ι)).

Examples
Here ι denotes the trivial representation, is the sign representation and σ is an irreducible 2-dimensional representation, for example, By choosing ρ 1 = σ on H 1 = C 2 and ρ 2 = ι + σ on H 2 = C 3 , we get a product system E → N 2 and an action of S 3 on O(E) O 2 ⊗ O 3 with fixed point algebra O(E) S 3 O ρ 1 ,ρ 2 isomorphic to a corner of the C * -algebra of a rank two graph Λ. The set of vertices is Λ 0 = {ι, , σ} and the edges are given by the incidence matrices This is because The isometric intertwiners are Here I π is the identity of H π for π ∈Ĝ and I i is the identity of H i for i = 1, 2. Since , , , , This data is enough to determine a rank two graph Λ associated to ρ 1 , ρ 2 . But this is not the only choice, since, for example, we could have taken which determines a different 2-graph. A direct analysis using the definitions shows that, in each case, the 2-graph Λ is cofinal, satisfies the aperiodicity condition and every vertex connects to a loop with an entrance. It follows that C * (Λ) is simple and purely infinite and the Doplicher-Roberts algebra O ρ 1 ,ρ 2 is Morita equivalent with C * (Λ).
The K-theory of C * (Λ) can be computed using Proposition 3.16 in [11] and it does not depend on the choice of factorization rules. We have In particular, O ρ 1 ,ρ 2 O 3 .
On the other hand, since ρ 1 , ρ 2 are faithful, both Doplicher-Roberts algebras O ρ 1 , O ρ 2 are simple and purely infinite with Even though ρ 1 , ρ 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 O ρ 1 ,ρ 2 is simple and purely infinite with trivial K-theory. In particular, O ρ 1 ,ρ 2 O 2 .
Note that, since ρ 1 , ρ 2 have kernel N = (123) Z/3Z, we could replace G by G/N Z/2Z and consider ρ 1 , ρ 2 as representations of Z/2Z. We haveĜ = {χ k : k ∈ Z}, where χ k (z) = z k and χ k ⊗ χ = χ k+ . The faithful representations ρ 1 = χ −1 + χ 0 , ρ 2 = χ 0 + χ 1 of T determine a product system E with O(E) O 2 ⊗ O 2 and a Doplicher-Roberts algebra O ρ 1 ,ρ 2 O(E) T isomorphic to a corner in the C * -algebra of a rank 2 graph Λ with Λ 0 =Ĝ and infinite incidence matrices, where The skeleton of Λ looks like and this 2-graph is cofinal, satisfies the aperiodicity condition and every vertex connects to a loop with an entrance, so C * (Λ) is simple and purely infinite. EXAMPLE 5.5. Let G = SU (2). It is known (see page 84 in [2]) that the elements inĜ are labeled by V n for n ≥ 0, where V 0 = ι is the trivial representation on C, V 1 is the standard representation of SU(2) on C 2 , and, for n ≥ 2, V n = S n V 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 = a b c d ∈ SU(2), (g · p)(z) = p(az 1 + cz 2 , bz 1 + dz 2 ).
If we choose ρ 1 = V 1 , ρ 2 = V 2 , then we get a product system E with O(E) O 2 ⊗ O 3 and a Doplicher-Roberts algebra O ρ 1 ,ρ 2 O(E) SU (2) isomorphic to a corner in the C * -algebra of a rank two graph with Λ 0 =Ĝ and edges given by the matrices 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, O ρ 1 ,ρ 2 is simple and purely infinite.