Spectra for compact quantum group coactions and crossed products

We present definitions of both Connes spectrum and strong Connes spectrum for actions of compact quantum groups on C*-algebras and obtain necessary and sufficient conditions for a crossed product to be a prime or a simple C*-algebra. Our results extend to the case of compact quantum actions the results in [8] which in turn, generalize results by Connes, Olesen and Pedersen and Kishimoto for abelian group actions. We prove in addition that the Connes spectra are closed under tensor products. These results are new for compact nonabelian groups as well.


Introduction
In his fundamental paper [4], Connes defines the invariant Γ called, in his name, the Connes spectrum, for abelian group actions on von Neumann algebras. Among other results, he obtained conditions for a crossed product to be a factor. Subsequently, Olesen and Pedersen [11] have defined the Connes spectrum for abelian group actions on C*-algebras. They have extended the results of Connes to the case of crossed products of C*-algebras by abelian group actions obtaining conditions that such a crossed product be a prime C*-algebra. However, the similar result for simple crossed products using the Olesen-Pedersen version of Connes spectrum is false. In [9], Kishimoto has shown that the result is true for simple crossed products if his "strong Connes spectrum" is used instead of Olesen and Pedersen ' Connes spectrum. Rieffel [15] and Landstad [10] have put the problem of finding a "good" definition of the Connes spectrum for compact, not necessarily abelian group actions on C*-algebras. In [10], Landstad remarks that a "good" definition of the Connes spectrum should lead to a result that generalizes the Olesen-Pedersen characterization of prime crossed products to the case of nonabelian compact group actions. Gootman, Lazar, Peligrad [8] have defined the Connes spectrum and the strong Connes spectrum for compact, not necessarily abelian group actions on C*algebras. In the case of abelian groups,these notions coincide with the previous ones. Moreover, in [8], Gootman, Lazar, and Peligrad prove the characterizations of the primeness and simplicity of crossed products using their definitions. In this paper we present definitions of both the Connes spectrum (Definition 3.1) and the strong Connes spectrum (Definition 4.1) in the case of compact quantum groups and prove the corresponding characterizations of primeness and simplicity of crossed products (Theorems 3. 4 and 4.4). In addition, we prove that the Connes spectra are closed under tensor products (Propositions 5. 4 and 5.6 ). This result is new for nonabelian compact groups as well. We use the techniques developed by Woronowicz [16,17], Boca [2], and the authors in [5,7]. In addition, this paper contains new methods for the study of the hereditary C*-subalgebras that are invariant under a compact quantum group coaction (Section 3) and for the proof of the key Lemma 2.2.

Preliminaries
Let G = (A, ∆) be a compact quantum group (see [17]) and let (B, G, δ) be a quantum dynamical system, where B is a C*-algebra and δ is a coaction of A on B (see [2] or [14]). Denote by G the set of all equivalence classes of irreducible representations of G ( [16], section 4).
representation of A called the adjoint of u α . We will denote by α the equivalence class of u α . Set We use F α to denote the unique positive, invertible operator in B(H α ), that intertwines u α with its double contragradient representation (u α ) cc such that tr(F α ) = tr(F −1 α ). Set M α = tr(F α ) ( [16], Theorem 5.4). Since every positive matrix is unitarily equivalent to a diagonal matrix, we may assume that the matrices F α are diagonal: F α = diag{f α 1 , . . . , f α dα }. The formula f α 1 (u α nm ) = δ nm f m defines a linear form on A. The above assumption implies that all u α ij are mutually orthogonal in H h and therefore where h is the Haar state on G and δ rs are the Kronecker δ's ( [16], Theorem 5.7).

SPECTRA FOR COMPACT QUANTUM GROUP COACTIONS AND CROSSED PRODUCTS 3
Define the mapping P α ij : Notice that B δ 2 (u α ) depends on the representative u α , not only on the equivalence class α ∈ G. However, for two equivalent representations u α 1 and u α 2 , the corresponding B δ 2 are spatially isomorphic. The proofs of the following remarks are straightforward from definitions.

Remark 2.1.
(1) If u α is a unitary representation of G, then , where 1 A is the unit of A and 1 B is the unit of the multiplier algebra of B. The leg numbering notation used here is the standard one ( [1] and [16]). Also, B δ 2 (u α ) is isomorphic as a Banach space to B δ α through the mapping X → dα i=1 P α ii (x). Therefore: is a matrix whose only non-zero row is the j 0 -row and whose j 0 jentry is given by P α i0j (x), for each j = 1, 2, . . . , d α . Furthermore, B δ 2 (u α ) = linspan{[P α ij (x rs )] ij | r, s = 1, 2, ..., d α }.
With α ∈ G and v the right regular representation of G, we will use the following notations (see [17] Denote by A the norm closure of the set of all operators of the form F v (a), where a ∈ A.
Recall that the crossed product B × δ G is defined to be the C*-algebra generated by all elements of the form (π u ×π h )(δ(b))(1⊗F v (a)), where a ∈ A, b ∈ B, π u : B → B(H u ) is the universal representation of the C*-algebra B and π h : A → B(H h ) is the GNS representation of A associated to the Haar state h.
Furthermore, if α 1 , α 2 ∈ G, define For properties of S α1,α2 see [5], Lemma 3.1 and Proposition 3.2. It is straightforward to check that Note that, from the above definition, an element ( , every such operator can be represented as a d 2 α × 1 column matrix with entries in B.
If v is the right regular representation, then , the C * -subalgebra of fixed points for the coaction ad(v): In [5], Section 2.3 it is noticed that, if u α is a representation of G on a Hilbert space H, the following is a coaction of G on B ⊗ K(H): [5], Section 4). The following result is a generalization of [13], Lemma 2.10 to the case of compact quantum groups. The proof uses the matricial representation of the elements of S α,ι discussed above. We will use this result in Section 3.
. Then, if η ∈ H u and ξ h is the cyclic vector in H h , for i 0 , j 0 = 1, 2, ..., d α we have: i0k and all the other entries are 0. Let now c ∈ B and c r0s0 = P α r0s0 (c). Then, similarly, (1 ⊗ p α )δ(c r0s0 )(1 ⊗ p ι ) can be represented by a 1 × d 2 α row matrix whose entry 1 × [(k − 1)d α + s 0 ] is c r0k and all the other entries are 0. Therefore, the product where each block X ij has the entry j 0 s 0 equal to b * i0i c r0j and the rest equal to 0, i.e.
Applying id ⊗ h to the above expression and using Formula 2 above, we get: Hence, if j 0 = s 0 , we have: is the matrix whose j 0 row is [cb i0i ] and the other entries are 0 and N ∈ B δ 2 (u α ) is the matrix whose s 0 = j 0 row is [c r0j ] and the other entries 0. If j 0 = s 0 then Q(X) = 0 but, as can be easily checked, also M * N = 0 and Ψ(M * N ) = 0.
Let now (B, G, δ) be a quantum dynamical system. We say that a C * -subalgebra C ⊂ B is δ-invariant if the following two conditions hold: The set of all hereditary, δ-invariant C * -subalgebras of B will be denoted by H δ (B).
A C*-algebra B is called G-prime if the product of two non-zero δ-invariant ideals is non-zero.
A C*-algebra B is called G-simple if B has no non-trivial δ-invariant two sided ideals.
We will need the following remarks. Their proofs are straightforward.

Remark 2.3.
(1) S ι = B δ ⊗ 1 (2) Using the proof of Proposition 3.2 in [5], one can show that for a 0 , a 1 ∈ B δ and α ∈ G, then The next lemma and its corollary will be used in Section 4.
and the claim follows.

Connes Spectrum and prime crossed products
A notion of spectrum of an action δ of a compact group G on a C * -algebra B was given in [8] by Gootman, Lazar, and Peligrad. They used the spectral subspaces B δ 2 (α) to define the Arveson and Connes spectra and proved that the conjugate α belongs to the Arveson spectrum Sp(δ) if and only if the closure of the ideal S α,ι * S ι,α is essential in S α (Proposition 1.3). We are going to use this correspondence rather than the direct definition given in [8] to define the spectra for coactions of a compact quantum group on a C * -algebra B.
The connection to the definition in [8] is made by the following lemma.
We will prove next the main result of this section. The result is a generalization of [8], Theorem 2.2.
Using similar arguments as in Lemma 3.2 we obtain the following result.
The following result makes a connection between the strong Connes spectrum and the simplicity of the fixed point algebra B δ .
Proof. Let J ⊆ B δ be a non-zero two sided ideal. We will prove that J = B δ and thus B δ is simple. To this end we will show that S ι,α (J ⊗ 1)S α,ι ⊆ J ⊗ 1, for any α ∈ G. The claim will then follow from Corollary 2.5.
We can now prove: Theorem 4.4. The following are equivalent: Proof. Assume first that B × δ G is simple. That B is G-simple follows easily since for every non-trivial ideal J ∈ H δ (B), J × δ G is a non-trivial ideal of B × δ G. Let now α ∈ G and C ∈ H δ (B). Then C × δ G si a hereditary subalgebra of B × δ G by Remark 2.3 (3) and hence it is simple. By [5] Corollary 4.9, S α,ι = 0 and S α is simple. Hence S α,ι S ι,α = S α and α ∈Γ(δ).

Spectra are closed under tensor products
In order to prove the results about the stability of the Connes spectrum and the strong Connes spectrum to tensor products, we need to make some notations. If α ∈ G and β ∈ G and u α ∈ α, u β ∈ β denote by u α ⊙ u β = p,q,r,s m α pq ⊗ m β rs ⊗ u α pq u β rs the Kronecker tensor product of u α and u β , which is a representation of A [17]. Then u α ⊙ u β is unitary if both u α and u β are unitary. Moreover, u α ⊙ u β is equivalent to a direct sum of irreducible representations, u α ⊙ u β = ⊕ i u ρi , ρ i ∈ G. The equivalence and ρ i ∈ G are unitary if both u α and u β are unitary [17].
Definition 5.1. Let Π ⊂ G be a subset. We say that Π is closed under tensor products if for every α ∈ Π, β ∈ Π and u α ∈ α, u β ∈ β it follows that every irreducible component of u α ⊙ u β belongs to Π.
(for the case of groups this notation was used in [12]). Standard calculations show that X ⊙ Y ∈ B δ 2 (u α ⊙ u β ). Furthermore, X ⊙ Y can be viewed as the matrix of order d α d β × d α d β partitioned in d 2 β blocks of order d α × d α as follows: with all the diagonal entries equal to Y ij and all the others equal to 0. (2)B δ 2 (u α ⊙ u β ) is spatially isomorphic to ⊕ B δ 2 (ρ i ). The proof of the above remark follows immediately using a change of basis in M dαd β .
We prove first Lemma 5.3.Sp(δ| C ) is closed under tensor products for every C ∈ H δ (B).
We will prove next that the Connes spectrum is closed under tensor products. As in the case of the strong Connes spectrum, we will show first that our Arveson spectrum Sp(δ| C ), is closed under tensor products for every C ∈ H δ (B).
Let α ∈ G and β ∈ G and u α ∈ α, u β ∈ β . If u α and u β are unitary, then, as noticed above, u α ⊙ u β is a unitary representation. If u α 1 is a representation in the class α, not necessarily unitary, then there exists an invertible matrix S ∈ M dα such that u α Proof. We may assume that C = B. Let α, β ∈ Sp(δ) and u α ∈ α, u β ∈ β be unitary representatives of α and β. We will first show that } is an essential ideal of (B ⊗ M dαd β ) δ u α ⊙u β . It then follows immediately that each irreducible component of u α ⊙ u β belongs to Sp(δ). Let . Let Z be partitioned in blocks as follows: , for every such X, Y . In particular, if Y is as in Remark 2.1 (4), that is Y has only one nonzero row consisting of y 1 , y 2 , ...y d β , we have for every X ∈ B δ 2 (u α ) and Y ∈ B δ 2 (u β ) as chosen, where the multiplication y i Z ij y * j is the multiplication in B of y i , y * j with each entry of Z ij . We prove first the following: for every Y ∈ B δ 2 (u β ) as chosen (i.e. with only one nonzero row). In the following leg numbering notation, there are four places in the following order: B, M dα , M d β , A.
Since Z ∈ (B ⊗ M dαd β ) δ u α ⊙u β , we have: By the definition of u α ⊙ u β , we have: On the other hand, taking into account that Y ∈ (B ⊗ M d β ) δ u β , it follows that: By combining Formulas 12 and 14 and taking into account that u α and u β are unitary, we get Formula 11. Therefore, since α ∈ Sp(δ) it follows that y i Z ij y * j = 0 for every such Y .
Let u α 1 ∈ α be a not necessarily unitary representation, but such that u α 1 is unitary. Then, since u α and u α 1 are equivalent, there is an invertible matrix S ∈ M dα such that u α Thus, since y i Z ij y * j = 0, it immediately follows that y i V ij y * j = 0, for every Y as chosen.
In particular, y i V pq ij y * j = 0, for all p, q = 1, 2...d α , where V pq ij is the entry pq of the d α × d α matrix V ij . Hence, we have Y DY * = 0. By Remark 2.1 (4) the matrices Y ∈ B δ 2 (u β ) that have only one non zero row, span B δ 2 (u β ) linearly. Therefore, Y DY * = 0 for every Y ∈ B δ 2 (u β ). Since Z ≥ 0 it follows that V ≥ 0 and so D ≥ 0. Therefore Y D = 0 for every Y ∈ B δ 2 (u β ). Notice that V = i,j V ij ⊗ m β ij satisfies Formula 12 with u α replaced by u α 1 . This fact will be used in the proof of the next Claim. Claim: The proof of the claim will be achieved in two steps: Step 1: We prove that Tedious but straightforward calculations show that the right hand side of the above formula is This last equality holds because we assumed that u α 1 is unitary. Therefore: and the Step 1 is proven.
Step 2: Proof of Claim. We have to prove that: We will evaluate separately the right and left hand sides of Formula 16 and show that they are the same. First, the right hand side: Next we will calculate the left hand side of Formula 16. As noticed above,by multiplying Formula 12 above by 1 B ⊗S * ⊗I d β ⊗1 A to the left and by 1 B ⊗S⊗I d β ⊗1 A to the right, and if we denote V = V ij ⊗ m β ij , we get δ 14 (V ) = (1 B ⊗ (u α 1 ⊙ u β ) * )(V ⊗ 1 A )(1 B ⊗ (u α 1 ⊙ u β )). Therefore: δ 14 (V ) = r,s,k,l,i,j,p,q,t,u,v,w Hence, if k = i 0 and w = j 0 we get: and if r = u = l, Therefore: Since u α 1 is a unitary representation, we have dα l=1 (u α 1 ) * pl (u α 1 ) ql = δ pq where, as usual, δ pq is the Kronecker symbol. Hence: Thus: Formulas 18 and 17 show that the Claim is true. Since β ∈ Sp(δ), D ∈ (B ⊗ M d β ) δ u β and Y D = 0 for every Y ∈ B δ 2 (u β ), it follows that D = 0. This means in particular that all the diagonal entries of the matrix V are equal to 0. Since V ≥ 0, it follows that V = 0 and thus Z = 0. Therefore linspan{(X ⊙ Y ) * (X ⊙ Y )|X ∈ B δ 2 (u α ), Y ∈ B δ 2 (u β )} is an essential ideal of (B ⊗ M dαd β ) δ u α ⊙u β as claimed.
We can now state: Proposition 5.6. The Connes spectrum, Γ(δ) is closed under tensor products.