THE RFD AND KAC QUOTIENTS OF THE HOPF * -ALGEBRAS OF UNIVERSAL ORTHOGONAL QUANTUM GROUPS

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THE RFD AND KAC QUOTIENTS OF THE HOPF * -ALGEBRAS OF UNIVERSAL ORTHOGONAL

QUANTUM GROUPS

Biswarup Das, Uwe Franz, Adam Skalski

To cite this version:

Biswarup Das, Uwe Franz, Adam Skalski. THE RFD AND KAC QUOTIENTS OF THE HOPF * -ALGEBRAS OF UNIVERSAL ORTHOGONAL QUANTUM GROUPS. 2020. �hal-03031501�

UNIVERSAL ORTHOGONAL QUANTUM GROUPS

BISWARUP DAS, UWE FRANZ, AND ADAM SKALSKI

Abstract. We determine the Kac quotient and the RFD (residually finite dimensional) quotient for the Hopf^∗-algebras associated to universal orthogonal quantum groups.

1. Introduction

Compact quantum groups of Woronowicz [Wor87] are often studied via their associated Hopf^∗-algebras, the so-called CQG algebras [DK94]. The CQG algebra carries all the group- theoretic information about the associated quantum group, such as its representation theory, the lattice of quantum subgroups (described via the lattice of the CQG quotients of the original algebra), or Kac property, but also for example encodes approximation properties of the natural operator algebraic completions.

When studying a particular property describing a ‘simpler’ class of objects, it is natural to ask whether a general object admits a largest subobject with the given property. And thus So ltan, motivated by the considerations concerning quantum group compactifications, showed in [So05] (see also [Tom07]) that every compact quantum group admits a unique maximal subgroup of Kac type; in other words, every CQG algebra admits a maximal Kac type quotient. He also computed such Kac quotients in some explicit examples, including the universal unitary quantum groups U_Q⁺ of Wang and Van Daele. The same paper also saw the first seeds of the study of residually finite dimensional CQG algebras, fully developed ten years later by Chirvasitu [Chi15]. The latter article shows that every CQG algebra admits the RFD quotient, which roughly speaking is the largest quotient which has ‘sufficiently many’

finite dimensional representations, discusses various stability results for the RFD property and most importantly proves that the CQG algebras of free unitary and orthogonal quantum groups,U_n⁺andO⁺_n are RFD for alln6= 3. The case ofn= 3 was established later in [Chi20].

One should note that already combining [So05], [Chi15] and [Chi20] leads to the description of the RFD quotient of the CQG algebras of all U_Q⁺. We also refer to these papers and their introduction for further motivation behind studying these concepts.

In this short note we compute the Kac and RFD quotients for the Hopf^∗-algebras associated to universal orthogonal quantum groupsO⁺_F of Wang and Van Daele, exploiting earlier results of Chirvasitu, the classification of O_F⁺up to isomorphism essentially due to Banica and Wang (formulated explicitly in [Rij07]), and the direct computations using the defining commutation relations. The main results are Theorems 3.3 and 3.4.

2. Preliminaries

We begin by recalling the basic objects and notions studied in this paper.

2020Mathematics Subject Classification. Primary 16T05, Secondary 20G42.

Key words and phrases. Hopf^∗-algebra; RFD property; Kac quotient; universal orthogonal quantum groups.

1

2.1. Universal compact quantum groups. We will study compact quantum groups in the sense of [Wor87] via the associated CQG (compact quantum group) algebras. These are involutive Hopf algebras which are spanned by the coefficients of their finite-dimensional unitary corepresentations, see, e.g., [KS97, Section 11.3]; each of them admits a unique bi- invariant state, called the Haar state. Note that Hopf^∗-quotients of CQG algebras are again CQG algebras, and the category of CQG algebras admits a natural free product construction (see for example [Wan95]).

The universal compact quantum groups U_Q⁺ and O⁺_F were introduced by Van Daele and Wang [WD96]. Let N ∈N, let F ∈M_N(C) be invertible, and putQ =F^∗F. The universal unitary CQG algebra Pol(U_Q⁺), also denoted A_u(Q), is generated by the N² coefficients of its fundamental corepresentation U = (u_jk)_1≤j,k≤N, subject to the conditions that U and F U F⁻¹ are unitaries inMN Pol(U_Q⁺)

. This means that for all 1≤j, k ≤N we have XN

ℓ=1

u_jℓu^∗_kℓ=δ_jk1 = XN ℓ=1

u^∗_ℓju_ℓk, (U1)

XN ℓ,r,s=1

u_ℓj(F^∗F)_ℓru^∗_rs(F^∗F)⁻¹_sk =δ_jk1 = XN ℓ,r,s=1

(F^∗F)_jru^∗_rs(F^∗F)⁻¹_rℓ u_kℓ. (U2)

Thus the CQG algebra Pol(U_F⁺∗F) depends only on the positive invertible matrix Q, which, up to isomorphism, we can assume to be diagonal, Q= (δ_jkq_j)_1≤j,k≤N, with 0< q1 ≤q2 ≤

· · · ≤q_N.

IfF satisfies furthermoreF F ∈RI_N, then we define the universal orthogonal CQG algebra Pol(O⁺_F), also denoted by B_u(F) or A_o(F), as the quotient of Pol(U_Q⁺) by the additional relation

(H) U =F U F⁻¹.

Up to isomorphism of CQG algebras, it is sufficient to consider the following two families, see [Wan02] and [Rij07, Remark 1.5.2].

Case I: F F =I_N, and F can be written as

(2.1) F =



 0 D 0 D⁻¹ 0 0

0 0 I_N−2k





with

D=



 q1

. ..

q_k





a diagonal matrix with coefficients 0< q1≤q2 ≤ · · · ≤q_k<1.

Case II: F F =−I_N,N is even, and F can be written as

(2.2) F =

0 D

−D⁻¹ 0

with

D=



 q₁

. ..

q_N/2





a diagonal matrix with coefficients 0< q₁≤q₂ ≤ · · · ≤q_N/2≤1.

Note that the eigenvalues of Q=F^∗F are given by

case I: 0< q²₁ ≤ · · · ≤q_k²<1< q_k⁻²≤ · · · ≤q⁻²₁ , case II: 0< q²₁ ≤ · · · ≤q_N/2² ≤1≤q_N/2⁻² ≤ · · · ≤q₁⁻², where in case I, 1 is an eigenvalue only if 2k < N.

2.2. Kac quotient and RFD quotient. IfA= Pol(G) is the CQG algebra of some compact quantum group, then the Kac ideal ofA is defined as the intersection of the (left) null spaces of all tracial states on A:

JKAC={a∈A;τ(a^∗a) = 0 for all tracial states τ on A},

and the Kac quotient isAKAC =A/JKAC. One can show thatAKACis again a CQG algebra, which corresponds to the largest quantum subgroup of G which is of Kac type; the last statement means that the associated Haar state is a trace.

So ltan [So05, Appendix A] [So06, Section 5] worked with the Kac quotient for C^∗-algebras associated with compact quantum groups, but here we prefer to use a version for CQG algebras, which is also the setting in [Chi15]. See Subsection 2.4 below for a brief discussion of the relation between CQG-algebraic and C^∗-algebraic Kac or RFD quotients.

Motivated by a question about Bohr compactifications of discrete quantum groups, Chir- vasitu introduced in [Chi15] the RFD property (where RFD stands for ‘residually finite di- mensional’) for CQG algebras and showed that Pol(U_N⁺) =Au(IN) and Pol(O⁺_N) =Bu(IN) = Ao(IN) have this property, implying that the discrete quantum groups Uc_N⁺and dO⁺_N are max- imal almost periodic in the sense of [So05, So06]. See also the related more recent paper [BBCW17].

The RFD quotient is defined as the biggest quotient of a CQG algebra that has the RFD property. We recall the relevant definitions from [Chi15].

Definition 2.1. [Chi15, Definition 2.6] A *-algebra A has property RFD, if for any a∈A, a 6= 0, there exists a finite-dimensional representation (i.e. a unital ^∗-homomorphism) π : A→M_n(C) with π(a)6= 0.

The RFD quotient ARFD of a *-algebra A is the quotient of A by the intersection of the kernels of all representations π:A→M_n(C), withn∈N.

In other words, ARFD =A/JRFD with

JRFD ={a∈A;∀π :A→M_n(C) a representation, π(a) = 0}.

One can show that the RFD quotient of a CQG algebra is again a CQG algebra.

Note that RFD is a weaker property than inner linearity (defined in [BB10], see also [BFS12]); in general the relationship between various possible notions of residual finiteness for quantum groups remains not fully clarified – see for example the comments in [BBCW17].

Chirvasitu proved the following three results.

Proposition 2.2. [Chi15, Last sentence of Section 2.4] If a CQG algebra has property RFD, then it is of Kac type.

Proposition 2.3. [Chi15, Proposition 2.10] If two *-algebras A and B have property RFD, then their free product A ⋆ B also has property RFD.

Theorem 2.4. [Chi15, Theorem 3.1], [Chi20, Theorem 2.4]The CQG algebras Pol(U_N⁺) and Pol(O⁺_N) have property RFD for N ≥2.

Remark 2.5. ForN = 1 we have Pol(U₁⁺) =CZand Pol(O⁺₁) =CZ₂, so property RFD also holds for N = 1, cf. [Chi15, Remark 3.2]. (More generally any commutative *-algebra that embeds into some C^∗-algebra has RFD, cf. [Chi15, Remark 2.7]).

The proofs in [Chi15] do not include N = 3; this case is dealt with in [Chi20].

The quotient CQG-algebras A_RFD and A_KAC yield quantum subgroups G_RFD and G_KAC of G. SinceJKAC⊆ JRFD, we haveG_RFD ⊆G_KAC, i.e.ARFD is a quotient ofAKAC.

2.3. RFD quotient of universal unitary quantum groups. The Kac quotients and the RFD quotients for the universal unitary quantum groups are already known, although the latter result has not been explicitly stated in the literature.

Theorem 2.6. [So05, Chi15, Chi20] Let Q ∈M_d(C) be an invertible positive matrix with r distinct eigenvalues q1, . . . , q_r, which have multiplicities M1, . . . , M_r.

Then the Kac quotient and the RFD quotient of the CQG algebra Pol(U_Q⁺) are equal to the free product ⋆^r_ν=1Pol(U_M⁺

ν).

Remark 2.7. So ltan showed that this is the Kac quotient, cf [So05, Theorem 4.9] and [So06, Section 7]. Chirvasitu’s results, i.e., Proposition 2.3 and Theorem 2.4, show that this free product is RFD, and therefore it is also the RFD quotient.

2.4. CQG-algebraic quotients vs. C^∗-algebraic quotients. Let G = (A,∆) be a com- pact quantum group with C^∗-algebraAand CQG algebraA. The C^∗-algebraic Kac ideal and RFD ideal are

J_KAC ={a∈A;τ(a^∗a) = 0 for all tracial states τ on A}, with J_KAC =A if Ahas no tracial states, and

J_RFD ={a∈A;π(a) = 0 for all fin.-dim. repr.π ofA}, with J_RFD =A if Ahas no finite-dimensional representations.

Again we can define A_KAC and A_RFD as respective quotients of A byJ_KAC and J_RFD, and again the RFD quotient is a quotient of the Kac quotient.

Since we can restrict tracial states or finite-dimensional representations ofAto A, we have JKAC ⊆J_KAC∩ A and JRFD ⊆J_RFD∩ A.

In general this inclusion can be proper. If A=C_u(G) is the universal C^∗-algebra of G, then we have equality, since every state and representation on Aextends to C_u(G).

Example 2.8. [CS19, Proposition 2.4] showed that a compact quantum group is coamenable if and only its reduced C^∗-algebra admits a finite-dimensional representation. Therefore, using the results of Banica from [Ban96], [Ban97] and [Ban99] we have C_r(U_Q⁺)RFD ={0}for N ≥2, andC_r(O⁺_F)_RFD ={0} forN ≥3.

Banica [Ban97, Theorem 3] showed also that the reduced C^∗algebra ofU_Q⁺admits a unique trace if Q ∈ RI_N, and no trace if this is not the case. Thus we get C_r(U_N⁺)KAC = C_r(U_N⁺) and C_r(U_Q⁺)_KAC ={0} if Q6∈RI forN ≥2. Similarly, ifN ≥3 and kFk⁸ ≤ ³₄TrF F^∗, then C_r(O⁺_F) has a unique trace ifF^∗F =I, and no trace ifF^∗F 6∈RI_N, see [VV07, Theorem 7.2].

Therefore we get in this case C_r(O⁺_N)_KAC =C_r(O_N⁺) andC_r(O⁺_F)_KAC ={0} if F^∗F 6∈RI_N. 3. RFD quotient of the universal orthogonal quantum groups

Let us now describe the RFD quotients of the free orthogonal quantum groups O_F⁺ intro- duced in the beginning of the last section.

3.1. Two special cases. Let us start with some special cases which will be useful in the next section when we treat the general situation.

Proposition 3.1. Let M ≥1 and let J_M be the standard symplectic matrix J_M =

0 I_M

−I_M 0

Then the CQG algebra Pol(O⁺_J

M) has property RFD.

Proof. ForM = 1, we haveO_J⁺₁ =SU(2) and the result is true (as the algebra in question is commutative, see Remark 2.5).

For the general case we can use the same proof as in [Chi15, Section 3].

Step 1: The natural analog of [Chi15, Proposition 3.3] holds. Denote by A^′ the unital *- subalgebra of A= Pol(U_2M⁺ ) generated by u^∗_jku_ℓm, 1≤j, k, ℓ, m ≤2M and by B^′ the unital

*-subalgebra of B = Pol(O_J⁺

M) generated by u^∗_jku_ℓm, 1≤j, k, ℓ, m ≤2M. Then there exists a unique CQG algebra isomorphismA^′ ∼=B^′ such that A^′ ∋u^∗_jku_ℓm 7→u^∗_jku_ℓm ∈B^′.

This isomorphism is simply the restriction to A^′ of the embedding of Pol(U_2M⁺ ) into CZ⋆Pol(O⁺_J

M) defined in [Ban97, Th´eor`eme 1 (iv)] by u_jk7→zu_jk, j, k = 1, . . . ,2M, where z denotes the generator ofZ viewed as an element ofCZ.

Step 2: The center of Pol(O_J⁺

M) is given by the morphism of CQG algebra γ : Pol(O_J⁺

M) → CZ₂ with γ(u_jk) = δ_jkt (where t denotes the generator of Z₂). The cocenter (i.e. the Hopf kernel of γ, see [Chi14, Definition 2.10]) is exactlyB^′. Indeed, γ is central, i.e., it satisfies

(γ⊗id)∆ = (γ⊗id)◦Σ◦∆ : Pol(O⁺_J

M)→CZ₂⊗Pol(O_J⁺

M)

where Σ denotes the flip, and any other central map can be factored throughγ. Furthermore, we have

B^′ = Hker(γ) ={b∈Pol(O_J⁺

M) : (γ⊗id)∆(b) = 1⊗b}.

Step 3: We can therefore apply [Chi15, Theorem 3.6] to prove an analogue of [Chi15, Propo- sition 3.8]: Pol(O_J⁺

M) is RFD if and only if Pol(U_2M⁺ ) is, and deduce from Theorem 2.6 above that Pol(O_J⁺

M) indeed has property RFD.

Let us consider next the case whereF^∗F has only two eigenvalues: q²<1< q⁻².

Proposition 3.2. Let M ∈N, q∈(0,1), ǫ∈ {−1,1} and set F =

0 qI_M ǫq⁻¹I_M 0

Then the RFD quotient and the Kac quotient of the CQG algebra Pol(O_F⁺) are both equal to the CQG algebra Pol(U_M⁺).

Proof. This proof is similar to those of Theorems 3.3 and 3.4 in the next subsection, therefore we will give a rather detailed argument here, and later sketch only the main steps. We decompose the fundamental corepresentationU as

U =

A B C D

, with

A= (u_jk)_1≤j,k≤M, B= (u_jk) 1≤j≤M M+1≤k≤2M

, C= (u_jk)M+1≤j≤2M 1≤k≤M

, D= (u_jk)_{M+1≤j,k≤2M} ∈M_M Pol O_F⁺)

. The defining relation (H) of Pol(O_F⁺) means that

U =F U F⁻¹ =

D ǫq²C ǫq⁻²B A

. So we can writeU as

U =

A ǫq²C

C A

, and therefore

U^∗=

A^∗ C^∗ ǫq²C^t A^t

. The unitarity condition forU now reads

AA^∗+q⁴CC^t AC^∗+ǫq²CA^t CA^∗+ǫq²AC^t CC^∗+AA^t

=

I_M 0 0 I_M

= (3.1)

=

A^∗A+C^∗C ǫq²A^∗C+C^∗A ǫq²C^tA+A^tC q⁴C^tC+A^tA

.

The equalities of upper left corners of (3.1) mean that for allj, k = 1, . . . , M XM

ℓ=1

(u_jℓu^∗_kℓ+q⁴u^∗_j+M,ℓu_k+M,ℓ) =δ_jk1 = XM ℓ=1

(u^∗_ℓju_ℓk+u^∗_ℓ+M,ju_ℓ+M,k).

Settingj=k and taking the sum, we get XM

j,ℓ=1

(u_jℓu^∗_jℓ−u^∗_jℓu_jℓ) = XM j,ℓ=1

(1−q⁴)u^∗_j+M,ℓu_j+M,ℓ. Letτ be a tracial state on Pol(O_F⁺). The equality above implies that

τ(u^∗_j+M,ℓu_j+M,ℓ) = 0

for allj, ℓ∈ {1, . . . , M}. So the generatorsu_jk withM+ 1≤j≤2M and 1≤k≤M, which form the matrix C, belong to the Kac idealJ_KAC.

If we divide by the *-ideal generated by the coefficients ofC, then we see from Equation (3.1) that the remaining generators u_jk with 1 ≤j, k ≤ M, which form the matrix A — or rather their images in the quotient *-algebra — have to satisfy exactly the defining relations of Pol(U_M⁺), i.e.,

AA^∗ =I_M =A^∗A and AA^t=I_M =A^tA.

The result now follows, since Chirvasitu proved that Pol(U_M⁺) is RFD, cf. Theorem 2.4.

3.2. Case I:F F =IN. We now look at the caseF F =IN, where we can assume that F has the form given in Equation (2.1). But we will permute the rows and columns ofF to organize F in blocks corresponding to the eigenvalues ofF^∗F.

Theorem 3.3. Let F be of the form

F =











0 q1I_M1

q⁻¹₁ I_M1 0

. ..

0 q_rI_Mr q_r⁻¹IMr 0

I_N−2K











with 0< q1 <· · ·< q_r<1 and K =M1+· · ·+M_r.

Then the RFD quotient and the Kac quotient of the CQG algebra Pol(O_F⁺) are both equal to the free product

⋆^r_ν=1Pol(U_M⁺_ν)

⋆Pol(O⁺_N−2K).

Proof. The proof is similar to that of Proposition 3.2.

Writing U as a block matrix and using the relation between the blocks that follow from (H), we can express U as

(3.2) U =















A₁₁ q²₁C₁₁ A₁₂ q₁q₂C₁₂ . . . R₁ C11 A11 C12 q1q₂⁻¹A12 . . . q⁻¹₁ R1

A21 q2q1C21 A22 q₂²C22 . . . R2

C21 q2q⁻¹₁ A21 C22 A22 . . . q⁻¹₂ R2

... ... ... ... . .. ... X₁ q₁X₁ X₂ q₂X₂ . . . Z













 ,

where furthermore the coefficients ofZ are hermitian, i.e.,Z =Z.

If we look at the diagonal blocks of the unitarity condition U^∗U =I_N =U U^∗, we get for everyµ= 1, . . . , r

Xr ρ=1

A^∗_ρµAρµ+C_ρµ^∗ Cρµ

+X_µ^∗Xµ=IMµ

= Xr ρ=1

A_µρA^∗_µρ+q²_µq_ρ²C_µρC_µρ^t

+R_µR^∗_µ, (3.3)

Xr ρ=1

q_µ²q²_ρC_ρµ^t Cρµ+q_ρ²q_µ⁻²A^t_ρµAρµ

+q²_µX_µ^tXµ=IMµ

= Xr ρ=1

CµρC_µρ^∗ +q_µ²q⁻²_ρ AµρA^t_µρ

+q_µ⁻²RµR^t_µ, (3.4)

Z^tZ+ Xr ρ=1

R^∗_µRµ+q_µ⁻²R^t_ρRρ

=IN−2K =ZZ^t+ Xr ρ=1

XρX_ρ^∗+q_ρ²XρX_ρ^t . (3.5)

Note that if

A= (a_jk)^1≤j≤J

1≤k≤K ∈M_J×K(A)

is a matrix with coefficients in some *-algebraA andτ is a tracial state on A, then we have τ ◦Tr(A^∗A) =

XJ j=1

XK k=1

τ(a^∗_jka_jk) =τ◦Tr(AA^∗) =τ ◦Tr(AA^t) =τ◦Tr(A^tA).

So ifτ is a tracial state on Pol(O⁺_F) and we apply τ◦Tr to Equation (3.5), then we get (3.6)

Xr ρ=1

(1 +q²_ρ)τ Tr(X_ρ^∗X_ρ)

= Xr ρ=1

(1 +q⁻²_ρ )τ Tr(R_ρ^∗R_ρ) .

If we now take the sum over µ of the difference between the left-hand-side and the right- hand-side in Equations (3.3) and (3.4), and applyτ ◦Tr, then we get

Xr ρ,µ=1

(1−q_µ²q_ρ²)τ Tr(C_ρµ^∗ C_ρµ) +

Xr µ=1

τ Tr(X_µ^∗X_µ)

− Xr µ=1

τ Tr(R^∗_µR_µ))

= 0, Xr

ρ,µ=1

(1−q_µ²q_ρ²)τ Tr(C_ρµ^∗ C_ρµ) +

Xr µ=1

q²_µτ Tr(X_µ^∗X_µ)

− Xr µ=1

q⁻²_µ τ Tr(R^∗_µR_µ)

= 0.

Adding these two relations and taking Equation (3.6) into account, we getτ Tr(C_ρµ^∗ C_ρµ)

= 0 for all ρ, µ ∈ {1, . . . , r}; by positivity this means that all the generators that appear in the C-blocks are contained in the Kac ideal JKAC.

By (3.3), we then also haveτ Tr(X_µ^∗)X_µ

=τ Tr(R^∗_µR_µ)

, so, plugging this into (3.6), Xr

ρ=1

(q_ρ⁻²−q_ρ²)τ Tr(X_ρ^∗X_ρ)

= 0,

and we getτ Tr(X_µ^∗X_µ)

= 0 = τ Tr(R^∗_µR_µ)

for µ= 1, . . . , r, since all terms in the above sum are non-negative. Once again using the fact that τ is positive we deduce that all the generators that appear in the X- and R-blocks are contained in the Kac idealJKAC.

Denote by

A= (A_ρµ)1≤ρ,µ≤r ∈M_K Pol(O_F⁺)

the matrix obtained fromU by deleting the generators in the even rows and columns in the block decomposition in Equation (3.2), as well as the last row and column.

If we divide the *-algebra Pol(O_F⁺) by the *-ideal generated by all C_ρµ, X_µ and R_µ, then unitarity relation U^∗U =I_N =U U^∗ reduces to

A^∗A=I_K =AA^∗ and DAD⁻¹A^t=I_K =A^tDAD⁻¹, Z =Z and ZZ^t=I_N−2K =Z^tZ,

where

D=



 q₁²I_M1

. ..

q_r²I_Mr



.

This means that the quotient Pol(O_F)/hC_ρµ, X_µ, R_µ : ρ, µ = 1, . . . , ri is equal to the free product of a copy of Pol(U_D⁺), generated by the coefficients of theA_ρµ,ρ, µ= 1, . . . , r, and a copy of Pol(O_N⁺_−2K), generated by the coefficients of Z.

Now we can conclude with Theorem 2.6.

3.3. Case II: F F =−I_N. Let us now consider the case F F =−I_N and F a matrix of the form given in Equation (2.2).

Theorem 3.4. Let N be an even positive integer and let F ∈M_N be of the form

F =









0 q1I_M1

−q₁⁻¹I_M1 0

. ..

0 q_rI_Mr

−q_r⁻¹I_Mr 0







 ,

with 0 < q1 <· · ·q_r−1 < q_r = 1, M1, . . . , M_r−1 ≥ 1, M_r ≥ 0, M1+· · ·+M_r = N/2. Note that M_r= 0 if 1 is not an eigenvalue of F^∗F.

The RFD quotient and the Kac quotient of the CQG algebraPol(O_F⁺) are both equal to the free product

⋆^r−1_ν=1Pol(U_M⁺

ν)

⋆Pol(O_J⁺), with

J =

0 I_Mr

−I_Mr 0

Proof. Like in the proofs of Proposition 3.2 and Theorem 3.3, we writeU as a block matrix.

Since U =F U F, we can write

U =













A11 −q²₁A11 A12 −q1q2C12 . . . A_1r −q1q_rC_1r C11 A11 C12 q⁻¹₁ q2A12 . . . C1r q⁻¹₁ qrA1r

A21 −q2q1C21 A22 −q²₂C22 . . . A_2r −q2q_rC_2r C₂₁ q₂⁻¹q₁A₂₁ C₂₂ A₂₂ . . . C_2r q⁻¹₂ q_rA_2r

... ... ... ... . .. ... ... A_r1 q_rq1C_r1 A_r2 −q2q2C_r2 . . . A_rr −q²_rC_rr C_r1 q_r⁻¹q1A_r1 C_r2 q⁻¹_r q2A_r2 . . . C_rr A_rr











 .

The unitarity conditions on the diagonal blocks read (ν = 1, . . . , r) Xr

µ=1

A^∗_µνAµν+C_µν^∗ Cµν

=IMν = Xr µ=1

AνµA^∗_νµ+q_µ²q_ν²CµνC_µν^t , (3.7)

Xr µ=1

q²_µq_ν²C_µν^t C_µν+q⁻²_µ q²_νA^t_µνA_µν

=I_Mν = Xr µ=1

C_νµC_νµ^∗ +q⁻²_ν q_µ²A_νµA^t_νµ . (3.8)

Letting τ be a tracial state on Pol(O_F⁺) and applying τ ◦Tr to the difference of the left- hand-side and the right-hand-side in Equation (3.7), we get

Xr ν,µ=1

(1−q_µ²q²_ν)τ Tr(C_µν^∗ Cµν)

= 0,

which implies that the coefficients appearing in all the C-blocks, except possibly C_rr, belong to the Kac ideal JKAC.

Taking now the differences of the left-hand-sides, or, respectively, right-hand-sides, in Equa- tions (3.7) and (3.8), we get

Xr−1 µ=1

(q⁻²_µ q²_ν−1)τ Tr(A^∗_µνA_µν)

= 0, Xr−1

µ=1

(1−q_µ²q⁻²_ν )τ Tr(A_νµA_νµ)^∗

= 0,

forν∈ {1, . . . , r−1}. From these two relations we can prove by induction thatτ Tr(A^∗_µνAµν)

= 0 for all µ, ν = 1, . . . , r−1 with µ6=ν.

Denote by J the *-ideal generated by the u_jk that have been regrouped in the blocks C_µν with (µ, ν) 6= (r, r), and in the blocks A_µν with µ 6= ν. It is not difficult to show that Pol(O⁺_F)/J ∼= ⋆^r_ν=1Pol(U_M⁺

ν)

⋆Pol(O_F⁺₀). As the latter CQG algebra is RFD by Chirvasitu’s results, it follows that this is indeed the RFD quotient and also the Kac quotient

of Pol(O_F⁺).

Acknowledgements

We acknowledge support by the French MAEDI and MENESR and by the Polish MNiSW through the Polonium programme. AS was partially supported by the NCN (National Science Centre) grant 2014/14/E/ST1/00525. UF was supported by the French ‘Investissements

d’Avenir’ program, project ISITE-BFC (contract ANR-15-IDEX-03), and by an ANR project (No. ANR-19-CE40-0002). We thank Anna Kula for several discussions on related topics.

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Instytut Matematyczny, Uniwersytet Wroc lawski, pl.Grunwaldzki 2/4, 50-384 Wroc law, Poland

Email address: biswarup.das@math.uni.wroc.pl

Laboratoire de math´ematiques de Besanc¸on, Universit´e de Bourgogne Franche-Comt´e, 16, route de Gray, 25 030 Besanc¸on cedex, France

Email address: uwe.franz@univ-fcomte.fr URL:http://lmb.univ-fcomte.fr/uwe-franz

Institute of Mathematics of the Polish Academy of Sciences, ul. ´Sniadeckich 8, 00–656 Warszawa, Poland

Email address: a.skalski@impan.pl