Gradient forms and strong solidity of free quantum groups

Consider the free orthogonal quantum groups ON+(F)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O_N^+(F)$$\end{document} and free unitary quantum groups UN+(F)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_N^+(F)$$\end{document} with N≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \ge 3$$\end{document}. In the case F=idN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F = \text {id}_N$$\end{document} it was proved both by Isono and Fima-Vergnioux that the associated finite von Neumann algebra L∞(ON+)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_\infty (O_N^+)$$\end{document} is strongly solid. Moreover, Isono obtains strong solidity also for L∞(UN+)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_\infty (U_N^+)$$\end{document} . In this paper we prove for general F∈GLN(C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F \in GL_N(\mathbb {C})$$\end{document} that the von Neumann algebras L∞(ON+(F))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_\infty (O_N^+(F))$$\end{document} and L∞(UN+(F))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_\infty (U_N^+(F))$$\end{document} are strongly solid. A crucial part in our proof is the study of coarse properties of gradient bimodules associated with Dirichlet forms on these algebras and constructions of derivations due to Cipriani–Sauvageot.


Introduction
In their fundamental paper [47] Ozawa and Popa gave a new method to show that the free group factors do not possess a Cartan subalgebra, a result that was obtained earlier by Voiculescu [66] using free entropy. To achieve this, Ozawa and Popa in fact proved a stronger result. They showed that the normalizer of any diffuse amenable von Neumann subalgebra of the free group factors, generates a von Neumann algebra that is again amenable. This property then became known as 'strong solidity'. As free group factors are non-amenable and strongly solid they in particular cannot contain Cartan subalgebras.
The approach of [47] splits into two important parts. The first is the notion of 'weak compactness'. [47] showed that if a von Neumann algebra has the CMAP, then the normalizer of an amenable von Neumann subalgebra acts by conjugation on the subalgebra in a weakly compact way. The second part consists in combining weak compactness with Popa's malleable deformation for the free groups and his spectral gap techniques. After the results of Ozwa-Popa several other strong solidity results have been obtained by combining weak compactness with different deformation techniques of (group-) von Neumann algebras, often coming from group geometric properties. Roughly (to the knowledge of the author) they can be divided into three categories: (I.1) The aforementioned malleble deformations; (I.2) The existence of proper cocycles and derivations and deformations introduced by Peterson [50] and further developed by Ozawa-Popa [48]; (I.3) The Akemann-Ostrand property, which compares to proper quasi-cocycles and bi-exactness of groups; c.f. [12,19,54]. For group von Neumann algebras the required property in (I.2) is to a certain extent stronger than (I. 3) in the sense that proper cocycles are in particular quasi-cocycles. These techniques have been applied successfully to obtain rigidity results for von Neumann algebras (in particular strong solidity results). The current paper also obtains such results and our global methods fall into category (I.2). Note also that we shall consider derivations on quantum groups without considering cocycles.
Recently, first examples of type III factors were given that are strongly solid [7], namely the free Araki-Woods factors. This strengthens the earlier results of Houdayer-Ricard [38] who showed already the absence of Cartan subalgebras. A crucial result obtained in [7] is the introduction of a proper notion of weak compactness for the stable normalizer of a von Neumann subalgebra. Using this notion of weak compactness strong solidity of free Araki-Woods factors is obtained by proving amenability properties of stable normalizers after passing to the continuous core.
This paper grew out of the question of whether the von Neumann algebras of (arbitrary) free orthogonal and free unitary quantum groups are strongly solid. These free orthogonal and unitary quantum groups have been defined by Wang and Van Daele [63] as operator algebraic quantum groups.
As C * -algebras the free orthogonal quantum groups are generated by self-adjoint operators u i, j , 1 ≤ i, j ≤ N with N ≥ 2 satisfying the relation that the matrix (u i, j ) 1≤i, j≤N is unitary. It was shown that this C * -algebra can be equipped with a natural structure of a C * -algebraic quantum group. Through a canonical GNS-construction this yields a von Neumann algebra L ∞ (O + N ). Parallel to this one may also define the free unitary quantum groups with von Neumann algebras L ∞ (U + N ), N ≥ 2. We refer to Sect. 2 below for details. These algebras have natural deformations parametrized by an invertible matrix F ∈ G L N (C) which yields quantum groups with non-tracial Haar weights (i.e. quantum groups that are not of Kac type). We write L ∞ (O + N (F)) and L ∞ (U + N (F)) for the associated von Neumann algebras. Ever since their introduction these algebras have received considerable attention and in particular over the last few years significant structural results have been obtained for them. In particular, recently it was proved that free quantum groups can be distinguished from the free group factors [11]. Further, the following is known if we assume N ≥ 3 (the case N = 2 corresponds to the amenable SU q (2) case): (1) Factoriality results for L ∞ (U + N (F)) and L ∞ (O + N (F)) were obtained in [24,64]. In particular for any F ∈ G L N (C) the von Neumann algebra L ∞ (U + N (F)) is a factor. If F = id N the factors are of type II 1 and otherwise they are of type III λ for suitable λ ∈ (0, 1]. (2) For a range of F ∈ G L n (C) the algebras L ∞ (O + N (F)) and L ∞ (U + N (F)) are non-amenable [4].
In the current context also the results by Voigt [65] on the Baum-Connes conjecture should be mentioned; part of the results of [24] and therefore the current paper are based on q-computations from [65].
In this paper we use quantum Markov semi-groups (i.e. semi-groups of state preserving normal ucp maps) and Dirichlet forms (i.e. their generators) to obtain strong solidity for all free orthogonal and unitary quantum groups. Dirichlet forms have been studied extensively [18,[20][21][22]29,35,58,60]. In particular in [21] it was shown that in the tracial case a Dirichlet form always leads to a derivation as a square root. The derivation takes values in a certain bimodule which we shall call the gradient bimodule. In this paper we show the following, yielding a H H + -type deformation as in [48,50] (see I.2 above): Key result (tracial case). Let G = O + N (F), F = Id N be the tracial free orthogonal quantum group. There is a Markov semi-group of central multipliers on G, which is naturally constructed from the results of [10,24], such that the associated gradient bimodule is weakly contained in the coarse bimodule of L ∞ (G).
In fact, the same result is true in the non-tracial case, but a stronger property is needed in order to treat that case by passing to the continuous core of a von Neumann algebra. The proof of the key result is based on two crucial estimates for the case F F ∈ RId N : one on the eigenvalues of the Dirichlet form and the other on intertwiners of irreducible representations of O + N (F) going back to [64]. In order to tackle all quantum groups U + N (F) and O + N (F) we treat the above in a more general context. We study semi-groups of state preserving ucp maps and introduce three properties: immediately gradient Hilbert-Schmidt (IGHS), gradient Hilbert-Schmidt (GHS) and gradient coarse (GC). IGHS (as well as GHS) essentially implies GC (see Proposition 4.4). The key result announced in the previous paragraphs is proved by showing that O + N (F) with F F ∈ RId N admits a semi-group that is IGHS. Preservation under free products and behavior under crossed products of IGHS and GC are studied in Sect. 5 from which we show that general free quantum groups admit semi-groups that are IGHS and their cores admit GC semi-groups.
These results suffice to fuel the theory as set out in the beginning of the introduction. We first recall the definition of strong solidity. We use the notion of weakly compact actions of stable normalizers from [7] and the deformation techniques (starting from proper derivations) as introduced by Peterson [50] and further developed by Ozawa-Popa [48]. Eventually this leads to strong solidity of all free orthogonal and unitary quantum groups. The precise statement we need from these sources does not occur in the literature (though very similar statements are claimed in [7,31,59]) and hence we incorporate them in the appendix.
We conclude: Note that if a Kac type quantum group with the CMAP has the Haagerup property then our approach here shows that there is a canonincal candidate for a bimodule (i.e. the gradient bimodule) and a proper real derivation into this bimodule. It remains then to show that the gradient bimodule is weakly contained in the coarse bimodule to obtain good deformations. It would be interesting to know how large the class of quantum groups is to which this strategy applies. Structure. Section 2 contains various preliminaries on quantum groups and von Neumann algebras. Section 3 recalls results by Cipriani-Sauvageot and some non-tracial extensions. Section 4 contains general results on Markov semi-groups and coarse properties of the gradient bimodule. Section 5 contains stability properties of IGHS, GHS and GC that are nedeed to treat O + N (F) for all F ∈ G L N (C). In Sects. 6 and 7 we prove our main theorem, i.e. the strong solidity result. Finally in Sect. 8 we prove a compression result. The parts that are directly taken from [7] and [48] are included in Appendix A.

Free orthogonal quantum groups
In [69] Woronowicz defined a compact C * -algebraic quantum group G = (A, A ) as a pair of a unital C * -algebra A with a comultiplication A : Compact quantum groups have a unique Haar state ϕ such that for x ∈ A, (2.1) Let (π ϕ , H ϕ ) be the GNS-representation of ϕ with cyclic vector ϕ := 1 ∈ H ϕ and set L ∞ (G) = π ϕ (A) . The state ϕ determines a unique normal faithful state, still denoted by ϕ, on L ∞ (G) satisfying (2.1) for all x ∈ L ∞ (G). Here G := L ∞ (G) is then the von Neumann algebraic comultiplication, which shall not be used in this paper. The triple (L ∞ (G), G , ϕ) is then a von Neumann algebraic quantum group in the Kustermans-Vaes sense, see [45]. It is common to write L 2 (G) for H ϕ . A finite dimensional unitary representation of G is a unitary element u ∈ L ∞ (G) ⊗ M n (C) such that ( G ⊗ id)(u) = u 13 u 23 with u 23 = 1 ⊗ u and u 13 = ( ⊗ id)(u 23 ) with the flip. We denote Irr(G) for the set of all irreducible representations modulo equivalence. For α ∈ Irr(G) we let u α be a corepresentation of class α; none of the constructions in this paper depend on the choice of the representative u α . We use α ⊆ β to say that α is a subrepresentation of β. This means that In the literature the terminology 'corepresentation' is also common to refer to representations, but here we stay with 'representation' as our terminology. Let α ∈ Irr(G) and let X α be the span of elements (id ⊗ ω)(u), ω ∈ M n (C) * and let H α = X α ϕ . X α is called the space of matrix coefficients of α. The projection of L 2 (G) := L 2 (L ∞ (G)) onto H α is denoted by p α and is called the isotypical projection of α.
We say that G is finitely generated if Irr(G) is finitely generated as a fusion category. That is, there exists a finite dimensional representation α such that for every β ∈ Irr(G) there exists a k ∈ N such that β ⊆ α ⊗k . We may assume that the trivial representation is contained in α and that α is equivalent to its contragredient representation. Then the minimal such k is called the length of β which we denote by l(β). The length depends on α, which at the point that we need it is implicitly fixed.
In [63] Wang and Van Daele introduced the free orthogonal quantum groups. We recall them here. Throughout the entire paper fix an integer N ≥ 2 and let F be a invertible complex matrix of size N × N . Let A := A(O + N (F)) be the universal C *algebra generated by elements u i, j , 1 ≤ i, j ≤ N subject to the relation that the matrix and Haar state ϕ. In case N = 2 the quantum group is amenable [4,9].
If we assume that F F ∈ RId N the quantum group O + N (F) is monoidally equivalent to SU q (2) where the number 0 < q < 1 is such that q + q −1 = Tr(F * F). Also set N q = q + q −1 which is the quantum dimension of the fundamental representation u 1 . It holds that N q ≥ N and equality holds if and only if the Haar state of O + N (F) is tracial. Note that q is the smallest root of x 2 − N q x + 1 = 0. In this case, i.e. when F F ∈ RId N , the representation theory of O + N (F) as a fusion category was described by Banica [4]. We have Irr(O + N (F)) N with 0 the trivial representation and 1 isomorphic to u 1 above. In fact we will denote u α for the representation of class α ∈ N. The fusion rules are for α ≥ β, We write n α for the dimension of α ∈ Irr(O + N (F)). It satisfies the recurrence relation N n α = n α+1 +n α−1 . If we let q 0 ∈ (0, 1) be the smallest positive root of x 2 −N x+1 = 0 then we have n α q −α 0 +O(1). Also q ≤ q 0 . It follows that lim sup α→∞ (n α ) 1/α q ≤ 1.

General von Neumann algebra theory
For von Neumann algebra theory we refer to the books by Takesaki [61,62].
Assumption Throughout the entire paper M is a von Neumann algebra with fixed normal faithful state ϕ. In case of a compact quantum group ϕ is the Haar state.
We use M op for the opposite von Neumann algebra and write x op , x ∈ M for elements in the opposite algebra. We also set x = (x * ) op . We write L 2 (M) for the standard form Hilbert space. It has distinguished vector ϕ such that x → x ϕ is a GNS-map for ϕ with ϕ(y * x) = x ϕ , y ϕ .

Tomita-Takesaki theory
Let S be the closure of the map We define the Tomita algebra T ϕ as the *algebra of x ∈ M that are analytic for σ ϕ . We write ξ x for J x * J ξ, ξ ∈ L 2 (M). Then where the latter set denotes the positive cone in the standard Hilbert space. We also record that [62,Lemma VIII.3.18],

Hilbert-Schmidt operators
Let H : Dom(H ) ⊆ M → M be a linear map. We say that H is Hilbert-Schmidt if the associated map L 2 (M) → L 2 (M) that sends x ϕ to H (x) ϕ is Hilbert-Schmidt. Wedenote the extension of H as a Hilbert-Schmidt map on L 2 (M) by H (l,2) . Then 2) e i , e j | 2 is the Hilbert-Schmidt norm for any choice of orthonormal basis e i . Let L 2 (M) be the conjugate Hilbert space of L 2 (M). We may identify H (l,2) isometrically and linearly with a vector ζ H ∈ L 2 (M) ⊗ L 2 (M) by means of the identification,

Bimodules and weak containment
An M-M-bimodule is a Hilbert space H with normal * -representations, π 1 of M and π 2 of the opposite algebra M op , that commute. Notation: aξ b = π 1 (a)π 2 (b)ξ with ξ ∈ H, a, b, ∈ M. We write M H M for the bimodule structure, or briefly H if the bimodule structure is clear. We recall the Connes-Jones definition of weak containment [26]. We also refer to the extensive treatment of bimodules by Popa [53]. Definition 2.1 Let K and H be two M-M-bimodules. We say that K is weakly contained in H, notation K H, if for every ξ ∈ K, ε > 0, E, F ⊆ M finite there exist finitely manyη j ∈ K indexed by j ∈ G such that for all x, ∈ E, y ∈ F, The following is Popa's definition of amenability [52,53].
This shows that ϕ ξ 1 ,ξ 2 (a ⊗b) = aξ 1 b, ξ 2 extends to a bounded functional on M⊗ min M op , moreover it is normal and thus extends to the von Neumann tensor product M ⊗ M op (by Kaplansky of the same norm). Take finitely many vectors ξ i of the above form and put ξ = i ξ i . As ϕ ξ,ξ is positive on M ⊗ min M op it extends to a positive normal functional on M ⊗ M op by Kaplansky. Then, as L 2 (M) ⊗ L 2 (M) is the standard form Hilbert space for M ⊗ M op , pick η ∈ L 2 (M) ⊗ L 2 (M) such that xξ y, ξ = xηy, η . As vectors ξ of this form are dense in H the lemma follows by approximation.

Quantum Dirichlet forms
Recall that a Markov map M → M was defined as a ϕ-preserving normal ucp map (the normal faithful state ϕ is always implicitly fixed and usually the Haar state of a compact quantum group in this paper). We say that a Markov map is ϕ-modular if x, y ∈ M.
If : M → M is any Markov map then by a standard interpolation argument there exists a contractive map (2) : KMS-symmetry is then equivalent to (2) being self-adjoint. With a Markov semigroup we mean asemi-group ( t ) t≥0 of KMS-symmetric Markov maps M → M such that for every x ∈ M the function t → t (x) is σ -weakly continuous. For ξ ∈ L 2 (M) we may write ξ = 3 k=0 i k ξ k with ξ k ∈ L + 2 (M) (the positive cone of the standard form). Let ξ + = ξ 0 .
A quadratic form Q is called conservative, completely Dirichlet if its matrix amplification Q [n] is a conservative Dirichlet form for every n ≥ 1. Here Dom(Q [n] ) are the n×n-matrices with entries in Dom(Q) and Q [n] . If Q is a quadratic form then write = Q ≥ 0 for the unique (closed densely defined) unbounded operator with Dom( 1 2 ) = Dom(Q) such that Q(ξ ) = 1 2 ξ, 1 2 ξ . The following result was obtained independently in [35] (in terms of Haagerup L p -spaces) and [20] (in terms of standard forms, being the formulation we take here).

Theorem 2.5 Q is a conservative completely Dirichlet form if and only if the semigroup (e −t ) t≥0 determines a Markov semi-group, meaning that there is a Markov
In view of Schönberg's correspondence [13,Appendix C], conservative completely Dirichlet forms are therefore non-commutative analogues of conditionally positive definite functions. We could have rephrased our statements in terms of conditionally negative definite functions by considering − instead of .
We need the following lifting property from [18,Lemma 5.2], the proof of which is essentially contained in [46]. We also recall that on the L 2 -level strong continuity and weak continuity of ( (2) t ) t≥0 are equivalent (see [15,Lemma 3.5]). σ -weak continuity of ( t ) t≥0 is equivalent to weak continuity of ( is called completely positive if Id n ⊗ T maps the positive cone in the standard form L + 2 (M n ⊗ M) into itself for every n ∈ N.
is a strongly continuous semi-group of self-adjoint completely positive operators on L 2 (M) such that S t ( ϕ ) = ϕ and such that S t (C) ⊆ C. Then there exists a Markov semi-group

Gradient forms and the results by Cipriani-Sauvageot
We recall some of the work of Cipriani-Sauvageot [21] which is crucial in our approach. We need a slightly more general version going beyond the case of tracial states of their construction. Note that we do not prove the existence of a square root in the non-tracial setting however (which is one of the main results of [21]; the question is also asked for in [60]).

The gradient bimodule
If Q is a conservative completely Dirichlet form on L 2 (M), then let ≥ 0 be such that Dom( Definition 3. 1 We assume that there is a σ -weakly dense unital * -subalgebra of the Tomita algebra T ϕ which we call A such that ∇ . That is, on the L 2 -and L ∞ -level agree under the symmetric correspondence. Finally, we assume that for every t ≥ 0 we have that t (A) ⊆ A and that ( t ) t≥0 is norm continuous on A. The latter implies that on the norm closure of A we have that ( t ) t≥0 is a C * -Markov semi-group and fits in the framework of [21]. 1 If ∇  3. 2 We note here that if ϕ is a trace τ then in [27] it was proved that Dom( 1 2 )∩ M is a * -algebra which may serve to do analogous constructions as we do below. We work with the algebra A however that is generally smaller in order to avoid some technicalities. In general we cannot guarantee the existence of such an algebra. Our assumption on the existence of A should also be compared to similar assumptions made in [43].

Remark 3.3
Suppose that G is a compact quantum group. Let A(G) be the * -algebra generated by all matrix coefficients u α i, j , α ∈ Irr(G). This algebra is well-known to be contained in the Tomita algebra T ϕ of the Haar state ϕ; in fact σ ϕ preserves the space spanned by u α i, j , 1 ≤ i, j ≤ n α for every single α ∈ Irr(G). Now if ( t ) t≥0 is moreover a semi-group of central multipliers, i.e. t (u α i, j ) = c α,t u α i, j for some constants c α,t ∈ C that form a semi-group, then it follows that A = A(G) satisfies the criteria described above. Indeed, in this case one has (u α i, j ) = c α u α i, j where c α is the derivative of c α,t at t = 0 from which this is directly derived. Definition 3.4 For x, y ∈ A we define the gradient form Quotienting out the degenerate part and taking the completion yields a Hilbert space H ∂ . The class of a ⊗ ξ will be denoted by a ⊗ ∂ ξ . We have contractive commuting actions (see below) determined by The proof of the following lemma is taken from the arguments in [21]. Since our setup is non-tracial and we work with the algebra A instead of the Dirichlet algebra of [21] we included a proof sketch.

Lemma 3.5 The operations (3.2) are (well-defined) contractive left and right actions of A that moreover commute.
Proof We first prove the statements for the left action. We need the auxiliary contractions A → M, We define an approximate gradient form by So that lim ε 0 x, y ,ε = x, y weakly in M. Exactly as in [21,Lemma 3.1] one proves that the approximate gradient form (3.3) is positive definite and that the Troughout the rest of the proof let And by the same argument backwards this yields therefore Contractiveness of the left action then follows by taking the limit ε 0. Next, for the right action we get So the right action is contractive. Clearly the left and right action commute.
Remark 3.6 By Lemma 3.5 the left and right action of A extend to the C * -closure of A. It is not clear to us whether these actions are normal in general and hence extend to actions on the von Neumann closure of A. However, in the bimodules that we require to prove our main theorem this is true, see Proposition 3.8 below.

Remark 3.7
Throughout the paper we shall often use the fact that for x, a, c ∈ A, ξ, η ∈ A ϕ we have,

Proposition 3.8 Let G be a finitely generated compact quantum group and let ( t ) t≥0 be a Markov semi-group of central multipliers. That is, for every t
Proof It suffices to show that the left and right action are σ -weakly continuous on the unit ball. Let a, b ∈ A(G) and assume moreover that they are coefficients of irreducible representations with length l(a) and l(b) respectively (see Sect. 2). Consider the mapping, c.f. (3.4), Now, let (x j ) j be a net in the unit ball of A(G) converging σ -weakly to x ∈ A(G). Take vectors ξ, η ∈ H ∂ arbitrary and let ε > 0. Let ξ 0 , η 0 be vectors in the linear span of all vectors a ⊗ ∂ b ϕ with a, b as above with ω ξ,η − ω ξ 0 ,η 0 ∂ < ε. By the previous paragraph and the polarization identity we may find j 0 such that for This shows that the left action is σ -weakly continuous on the unit ball. For the right action the proof is similar.

Derivations in the tracial case
The constructions of Sect. 3 work for non-tracial ϕ. Now assume ϕ is tracial, say ϕ = τ . Consider the linear map (3.6) Because in the tracial case a τ = τ a, a ∈ A we have for a, b ∈ A, i.e. ∂ is a derivation. Moreover, as by conservativity of we have τ ( (a * a)) = (a * a) τ , τ = a * a τ , τ = 0 and we see that, In [21,Section 4] it is proved that there exists a closable derivation ∂ 0 : Dom( . By construction ∂ ⊆ ∂ 0 and so ∂ is preclosed and we let ∂ be its closure. If A τ is a core for 1 2 it follows from (3.7) that the Dom(∂) contains the Dirichlet algebra of all x ∈ M such that x τ ∈ Dom( 1 2 ). So if A is a core for 1 2 then the derivation ∂ equals the closure of the derivation ∂ 0 constructed in [21,Section 4].
In the cases we need it these conditions are satisfied.

Lemma 3.9 Let ( t ) t≥0 be a semi-group of central multipliers on a compact quantum group G of Kac type (i.e. with tracial Haar state). Let be the generator of (
Proof Let p α be the isotypical projection of L 2 (G) onto the space of matrix coefficients of α ∈ Irr(G). As ( t ) t≥0 are central multipliers there exist constants α such that p α ξ = α p α ξ for any ξ ∈ L 2 (G). Let ξ ∈ Dom( 1 2 ). Then taking limits over increasing finite subsets F ⊆ Irr(G) we find α∈F p α ξ → ξ and α∈F p α .

Coarse properties of the gradient bimodule: IGHS, GHS and GC
In this section we study when the bimodule H ∂ is weakly contained in the coarse bimodule. We use all notation introduced in Sects. 2 and 3. In particular M is a von Neumann algebra with fixed normal faithful state ϕ. We let ( t ) t≥0 be a Markov semigroup on M and associate to it the generator , the algebra A, the Dirichlet form Q and the gradient form , . As A is contained in M it inherits the matrix norms of M and therefore complete positivity of a map A → M is understood naturally as a map that sends positive operators to positive operators on each matrix level. We introduce three properties of semi-groups that are convenient in studying coarse properties of H ∂ .

Definition 4.1
We call a Markov semi-group ( t ) t≥0 on a von Neumann algebra M with fixed normal faithful state ϕ immediately gradient Hilbert-Schmidt (IGHS) if for every choice a, b ∈ A we have that the following two properties hold: Remark 4. 2 We shall often make use of the fact that for a, b, x ∈ A,

Then the mapping
is Hilbert-Schmidt and converges strongly to 0 as t 0 (resp. for t = 0 the map (4.4) is Hilbert-Schmidt).
Proof The fact that for any choice of the x, a i , shows that • 0 is positive and the same argument on matrix levels gives complete positivity. Hence as t is completely positive also (4.3) must be completely positive.
Further, recalling t := a 1 ,...,a n ;c 1 ,...,c n t , n k,l=1 so that (4.5) is positive. The final statement follows as if the semi-group is IGHS, then is Hilbert-Schmidt for t > 0 by linearity and bounded if t = 0. Further t → 0 strongly as t 0. The statement for GHS follows similarly.

Proposition 4.4 Assume that the left and right A-actions on H ∂ extend to normal M-actions. If ( t ) t≥0 is IGHS or GHS then it is GC.
Proof We give the proof for the IGHS assumption; for the GHS assumption the proof is similar and in fact easier. Throughout the proof fix a 1 , . . . , a n , c 1 , . . . , c n ∈ A and for t ≥ 0 let t := a 1 ,...,a n ,c 1 ,...,c n t be the map defined in (4.4).
If x, y ∈ M are arbitrary we may approximate them using Kaplansky's density theorem in the strong topology with bounded nets (x k ) k and (y k ) k in A. Then x k → x in the σ -weak topology and x k ϕ → x ϕ in the norm of L 2 (M). Similarly y k → y σ -weakly and ϕ y * k = J y k ϕ → J y ϕ = ϕ y * in norm. The left and right Maction on H ∂ are normal and the IGHS assumption gives that is bounded In turn we find by the IGHS assumption that for all x, y ∈ M, By the IGHS assumption for t > 0 the map t is bounded L 2 (M) → L 2 (M) and moreover Hilbert-Schmidt by Lemma 4.3 and therefore we see that there exists a vector ζ t ∈ L 2 (M) ⊗ L 2 (M) such that, This shows that for t > 0 we have that Therefore, for every x, y ∈ M we have We can now directly check that H ∂ is weakly contained in the coarse bimodule of Then by approximation we find that for general ξ ∈ H ∂ we can find t > 0 such that for all x, y ∈ F the estimate (4.8) holds. We see by Definition 2.1 that H ∂ is weakly contained in the coarse bimodule of M.

Stability properties
We prove that IGHS and GHS are properties that are preserved by free products. We also prove the necessary reduction to continuous cores.

Free products
For the definition of free products of von Neumann algebras we refer to [2,66]. We also refer to [17] and adopt its notation and terminology. Let (M i , ϕ i ), i ∈ I be von Neumann algebras with normal faithful states ϕ i . The free product (M, ϕ) is the von Neumann algebra with normal faithful state ϕ that contains each M i , i ∈ I as a subalgebra to which ϕ restricts as ϕ i ; moreover, these algebras are freely independent in M with respect to ϕ and generate M.
it is Markov with respect to ϕ i ) then there exists a unique normal ϕ-preserving ucp map * i∈I i on the free product (M, ϕ) such that for a reduced operator Let i be the generator of ( i,t ) t≥0 and let A i be the dense unital subalgebras in M i as described in Sect. 2. Let be the generator of ( t ) t≥0 . Let a 1 . . . a n be a reduced operator of type A in the algebraic free product A = * i∈I A i . Then by taking σ -weak limits (which exists on these reduced operators), we obtain the following Leibniz rule, . . a n . (5.1) A rather tedious computation purely based on this Leibniz rule now shows the following. t≥0 and let ( t ) t≥0 be the free product Markov semi-group on the free product Proof The proof splits in steps. 1. Setup: expansion into reduced words. Let A i and A = * i A i as in the paragraph before this proposition. In particular the unit is in We assume moreover that all letters a i , b i and x i come from ∪ j O j . Let A, B and X be the respective types of a, b and x. To reduce the number of cases we need to consider in this proof we extend our notation as introduced above a bit. We shall write In particular, if x ∈ O i and y ∈ O j then • x y = x y if i = j (this extends the notation). In case i = j we have (x y) = (x)y + x (y) by the Leibniz rule (5.1). If we start writing bxa as a sum of reduced operators we find the following terms, where we define I and II as the big sums. We use the convention that a j = 0 if j > m and b j = 0 if j > k. Also note that many of these terms are 0, for example if The summands in I are reduced operators, the summands in II are not necessarily reduced for the reason that b k−i+1 x i a n−i+1 is not necessarily reduced. In order to treat this summand we continue our expansion into three sums and a remainder part F(x). We find that, where F : M → M is the finite rank operator that collects the remaining terms of II; that is, F(x) is given by the same expression (5.3) but with the operation • · replaced by taking ϕ( · ). 2. Appyling the -map. Now we apply a,b * t for t = 0 to x (we prefer a,b * t over a,b t to keep the notation simpler; for the proof it is irrelevant). Recall that, We proceed by expanding the right hand side of this expression into a decomposition very similar to (5.2) and (5.3). If we do this we get the following, where the respective terms II b (xa) , II (bxa) , II b (x)a and II (bx)a are described below. Write l k for if k = l and for the identity operator otherwise. So, . . . ϕ( Therefore, as all these terms cancel, For the 'II-terms' we get the following. Again, we split this into a decomposition similar to (5.3). We get that + F b,a,3 (x), where the F b,a,i 's are finite rank maps M → M and the II (1) , II (2) and II (3) terms are specified below. Let us first examine the II (1) -terms. We get that, Again we see that, (bx)a = 0. Now for the II (2) -terms we find, Again we get (or in fact by a symmetry argument from the II (1) -case), We now examine the II (3) -terms. We find, . . . ϕ(x i+1 a n−i ) is a finite rank map. Similarly, there are finite rank maps M → M, say G b,a,2 , G b,a,3 and G b,a,4 (in fact G b,a,3 being the 0 map as i+1 a n−i+2 . . . a m  + G b,a,4 (x). (5.6) As (1) = 1 (by conservativity of the Dirichlet form) we have for any y that (y) = ( • y ). We see that the first summations of the 4 terms of and (5.6) cancel each other, so that we get a remaining term: Now if we collect all of the above terms we see that with F a,b the finite rank operator F a, b = (F b,a,1 − F b,a,2 − F b,a,3 + F b,a,4 ) + (G b,a,1 − G b,a,2 − G b,a,3 + G b,a,4 ).

Conclusion of the proof.
Let F a,b H S be the Hilbert-Schmidt norm of F a,b as a map y ϕ → F a,b (y) ϕ . Now note that if the length n of x as a reduced operator is strictly longer than k + m − 1 then the expression (5.7) is 0 as there must be an operator b k+1 or a m+1 occuring in (5.7) which by definition are 0.

If each
. So we conclude that the second bullet of Definition 4.1 holds for the free product semi-group ( t ) t≥0 if it holds for each individual ( i,t ) t≥0 . It remains to verify the first bullet point of Definition 4.1.
Set E as the set of all reduced operators of the form e i 1 . . . e i n with e i n ∈ ∪ j O j . E forms an orthonormal basis of L 2 (M • ) = L 2 (M) C ϕ . Fix t > 0 and let C = max j σ i/2 ( t (a j )) and then C = max(1, C ). Further set D = max j t (b j ) and then D = max(1, D ). We conclude from (5.7) and twice Cauchy-Schwarz that, For all j we have because O B j is an orthonormal basis. Similarly, which is finite for every t > 0 and for every choice of a and b in A. The proof for GHS instead of IGHS follows just by using t = 0 instead of t > 0.

Crossed product extensions
We prove that IGHS semi-groups yield GC semi-groups on their continuous cores. We recall the following from [62]. As before let M be a σ -finite von Neumann algebra with fixed faithful normal state ϕ. Let c ϕ (M) be the continuous core von Neumann algebra of M. It is the von Neumann algebra acting on L 2 (M) ⊗ L 2 (R) L 2 (R, L 2 (M)) that is generated by the operators and the shifts We shall write u f = R f (s)u s ds for f ∈ L 1 (R). The map π ϕ embeds M into c ϕ (M). We let L ϕ (R) be the von Neumann algebra generated by u t , t ∈ R. Let ϕ be the dual weight on c ϕ (M) of ϕ. If s → x s and s → y s are compactly supported σ -weakly continuous functions R → M, it satisfies We call the support of s → x s the frequency support of R π ϕ (x s )u s ds. Let h ≥ 0 be the self-adjoint operator affiliated with L ϕ (R) such that h it = u t , t ∈ R. There exists a normal, faithful, semi-finite trace τ on c ϕ (M) such that we have cocycle derivative (D ϕ/D τ ) t = h it . This is informally expressed as τ (h 1/2 · h 1/2 ) = ϕ( · ). We write For x ∈ n ϕ we write x ϕ for its GNS-embedding into L 2 (c ϕ (M), ϕ). Let J ϕ be the modular conjugation.
The Tomita algebra T ϕ is defined as the algebra of all x ∈ c ϕ (M) that are analytic for σ ϕ and such that for every z ∈ C we have σ ϕ z (x) ∈ n ϕ ∩ n * ϕ . It shall be convenient for us to identify unitarily Remark 5. 2 We may similarly set For x ∈ n τ we writex τ for its GNS-embedding into L 2 (c ϕ (M), τ ). We have the c ϕ (M)-c ϕ (M)-bimodule structure given by x · (a τ ) · y = x J τ y * J τ (a τ ), a ∈ n τ , x, y ∈ c ϕ (M).

Consider the map
where D ⊆ L 2 (c ϕ (M), ϕ) is the space of x ∈ n ϕ such that xh Recall that a Markov semi-group If is a ϕ-modular Markov semigroup then so is for both the weights ϕ and τ , meaning that it is a point-strongly continuous semi-group of ucp maps that preserves these weights. If p ∈ L ϕ (R) is a τ -finite projection then the restriction of to the corner pc ϕ (M) p is a Markov-semigroup with respect to τ . The convention is mainly made to simplify several technicalities occuring in the proofs of Lemmas 5.4, 5.7 as well as Proposition 5.8. Let ( t ) t≥0 be a Markov semigroup of central multipliers. Let ≥ 0 be a generator for ( t ) t≥0 , i.e. e −t = (2) t . Let p ∈ L ϕ (R) be a projection. Then ⊗ p is a generator for the restriction of ( (2) t ) t≥0 to pc ϕ (M) p. Its domain is understood as all 2 -sums exists as a 2convergent sum.

Definition 5.3
Let A be the * -algebra of elements R π ϕ (x s )u s ds ∈ c ϕ (M) with x s ∈ A σ -weakly continuous and compactly supported in s.

Lemma 5.4 Let ( t ) t≥0 be a Markov semi-group of central multipliers on a compact quantum group G. Let A = A(G) and let A be defined as above. Then A is contained in the Tomita algebra T ϕ and moreover (∇
Further,we may set (the limit being existent), Moreover, Proof The inclusion A ⊆ T ϕ follows from the explicit form of the modular group of ϕ, see [62,Theorem X.1.17]. If s → x s ∈ A is continuous and compactly supported, it takes values in the space of matrix coefficients of a single finite dimensional representation of G. Write x s = α x s,α where α ranges over this finite (s-independent) subset of Irr(G). Then (π ϕ (x s )) = α α π ϕ (x s,α ) for some α ∈ C. Further, for And further, Just as in the state case this defines an inner product on A ⊗ L 2 (c ϕ (M)). Quotienting out the degenerate part and taking a completion yields a Hilbert space H ∂,c ϕ with contractive left and right A-actions given by We also set the map

Proposition 5.8 Let ( t ) t≥0 be a Markov semi-group of central multipliers on a compact quantum group G. Let p ∈ L ϕ (R) be a projection. Then: (1) The A-A-bimodule H ∂,c ϕ extends to a normal c ϕ (M)-c ϕ (M)-bimodule. Moreover, pH ∂,c ϕ p is a normal pc ϕ (M) p-pc ϕ (M) p-bimodule. (2) If ( t ) t≥0 on (M, ϕ) is IGHS then the Markov semi-group (c ϕ ( t )) t≥0 on pc ϕ (M) p is GC.
Proof To keep the notation simple we will identify A as a subalgebra of c ϕ (M) through the embedding π ϕ and further supress π ϕ in the notation. We prove the statements for the projection p = 1 and then justify how the general statements follow from this. Throughout the entire proof let f 1 , f 2 , g 1 , g 2 ∈ C 00 (R), a, b, c, d ∈ A. Proof of (1) for p = 1. Let x ∈ A. We have, We have (5.11) Now to show that the left A-action on H ∂,c ϕ is normal it suffices to show that it is σ -weakly continuous on the unit ball of c ϕ (M). So suppose that x k is a net in the unit ball of A converging σ -weakly to x.We get that a,c (x k ) ∈ A and may be written as a,c (x k ) = π ϕ (y k,s )u s ds, with integral ranging over some compact set. Similarly write a,c (x) = π ϕ (y s )u s ds. Let u α i, j be a matrix coefficient of α ∈ Irr(G). Then ( y k,s ϕ , u α i, j ϕ ) s∈R is an element of L ∞ (R) that is σ -weakly convergent to ( y s ϕ , u α i, j ϕ ) s∈R . It follows then from the expression (5.11) that Since x k is bounded it follows that for ξ ∈ H ∂,c ϕ arbitrary we get that (x −x k )ξ, ξ → 0. This concludes the claim on the left action; the right action goes similarly. Proof of (2) for p = 1. Assume that ( t ) t≥0 is IGHS. For x ∈ A we have as in (5.10) We argue that in fact (5.12) holds for all x, y ∈ c ϕ (M). Indeed, A is strong- * dense in c ϕ (M) so by Kaplansky's density theorem we may take bounded nets x k and y k in A converging in the strong- * topology (hence σ -weakly) to x ∈ c ϕ (M) and y ∈ c ϕ (M) respectively. By Step 1 the left and right actions are normal (meaning σ -weakly continuous) so that Since a,c is bounded by Lemma 5.7 andx k u f 1 ϕ → xu f 1 ϕ in norm we find 14) The limits (5.13) and (5.14) show that (5.12) holds for all x, y ∈ c ϕ (M). Further, by strong continuity of the semi-group we find for all x, y ∈ c ϕ (M), By the unitary identification (5.8) there exist z s , z s ∈ M such that, We have We may express the limiting terms on the right hand side of (5.15) as follows by using Lemma 5.7, a,c It follows in particular that for all s, r ∈ R we have where the symmetric support is defined as Then ζ a,c t defines an element of L 2 (R 2 , M ⊗ M) L 2 (c ϕ (M)) ⊗ L 2 (c ϕ (M)).
Note that (z s ) s∈R , hence ( σ ϕ s (a),c t (z s )) s∈R , is supported on the product of the sets ssup( f 1 ), ssup( f 2 ) and [−2n, 2n]. Hence given by The range of R n is contained in the elements with frequency support in [−2n, 2n]. Fix n and put K as before (5.18). It follows from (5.19) that for t > 0 the inner product functional  We can now conclude the proof as follows. Now let ε > 0 and let F be a finite subset in the unit ball of c ϕ (M). Let ξ = i u f i a i ⊗ b i u g i ϕ where the sum is finite and a, b ∈ A, f i , g i ∈ C 00 (R). Since R n → Id c ϕ (M) in the point-σ -weak topology we may take n ∈ N large such that for all x, y ∈ F we have | xξ y, ξ ∂ − R n (x)ξ y, ξ ∂ | < ε. (5.22) Recall from (5.12) that ω n,0 (x ⊗ y op ) = R n (x)ξ y, ξ ∂ . We may take t > 0 small such that for all x, y ∈ F, Combining (5.21), (5.22), (5.23) we find that for all x, y ∈ F we have | xξ y, ξ ∂ − xη n,t y, η n,t | < 2ε.
As the vectors ξ as above are dense in H ∂,c ϕ it follows that the bimodule H ∂,c ϕ is weakly contained in the coarse bimodule of c ϕ (M).
Then we see that we have a weak containment of the pc ϕ (M) p-pc ϕ (M) p-bimodules pH c ϕ ,∂ p in pL 2 (c ϕ (M)) ⊗ L 2 (c ϕ (M)) p. The latter is in turn weakly contained in pL 2 (c ϕ (M)) p ⊗ pL 2 (c ϕ (M)) p, which is justified by the following. If c ϕ (M) were to be a factor we write 1 = ∨ n p n with p n projections with τ ( p n ) = τ ( p); by comparison of projections there are unitaries u n such that u * n u n = p n and u n u * n = p. Then ξ → ξ u n (resp. ξ → u * n ξ ) intertwines the left (resp. right) action of c ϕ (M) on L 2 (c ϕ (M)) p and L 2 (c ϕ (M)) p n (resp. pL 2 (c ϕ (M)) and p n L 2 (c ϕ (M))). From this the weak containment follows in the factorial case. In general it follows from desintegration to factors of c ϕ (M).

The quantum group O + N (F) with FF ∈ RId N admits an IGHS Markov semi-group
In this section we make an analysis of semi-groups associated with L ∞ (O + N (F)) and its associated gradient bimodule. The idea is based on results from [24] where De Commer, Freslon and Yamashita have obtained the Haagerup property for O + N (F). We use general results from [18,42] to construct a semi-group for such O + N (F) that is IGHS.

Semi-groups and Dirichlet forms, case FF ∈ Rid
Let U α be the derivative of U α . The arguments in the proof of Proposition 6.2 below are close to constructions from [42,58] and its non-tracial generalization [18,Proposition 5.5]. We use these ideas to obtain a specific generator of a Markov semi-group that can be expressed in terms of the Chebyshev polynomials.
We need the fact that if P is a function that is smooth in a neighbourhood of 0 then, Recall that throughout the entire paper we made the convention that 0 < q ≤ 1 is fixed by the property q + q −1 = Tr(F * F) = N q . Define, where N q = q + q −1 is the quantum dimension of the fundamental representation of O + N (F). Proof This is shown in [31,Lemma 4.4] (in fact it can be derived rather directly from the recursion relation of U α ). [23,Theorem 17] it was proved that for every −1 < t < 1 we have that,

Proposition 6.2 Assume that F F ∈ RId N . There exists a Markov semi-group
determines a normal unital completely positive multiplier on L ∞ (O + N (F)). Note that the maps ϒ t with −1 < t < 1 mutually commute. Moreover, for . The proof of [18,Proposition 5.5] argues that we may define semi-groups of completely positive contractions S t,k = exp(−tγ k ) on L 2 (M). Further, Taking the limit k → ∞ of this expression and using (6.1) gives lim k→∞ S t,k (u α i, j ) = exp(−tc α (1)) u α i, j . By density we may conclude that for every ξ ∈ L 2 (O + N (F)) we have that S t,k (ξ ) is convergent say to S t (ξ ). Furthermore (S t ) t≥0 is a semi-group that is moreover strongly continuous (again this follows by comparing actions on A(O + N (F)) in L 2 (O + N (F)) and then using density). Consider the closed convex sets in L 2 (O + N (F)) given by C 0 = {x ∈ L 2 (M) | 0 ≤ x ≤ ϕ } and the positive cone in the i-th matrix amplification C i = M i (L 2 (M)) + where i ∈ N ≥1 . As for each t, n and i we have S t,n (C i ) ⊆ C i we get S t (C i ) ⊆ C i . Further S t ( ϕ ) = ϕ . Lemma 2.6 then shows that there exists a Markov semi-group we see that So the proposition follows by scaling the generator of the semi-group (S t ) t≥0 .
The following is now another example of [18, Theorem 6.7].

Corollary 6.3 Assume that F F ∈ RId N . There exists a conservative completely Dirichlet form Q N associated with O
Here α is defined in (6.2). Proof This is a direct consequence of the correspondence between conservative Dirichlet forms and Markov semi-groups, see Sect. 2.6 and [18, Section 6].

Properties IGHS and GHS
We prove that the Markov semi-group constructed in Proposition 6.2 is IGHS and even GHS in the non-tracial case.
where stands for an inequality that holds up to a constant that does not depend on α, β and γ .
Proof For each m, n ∈ Z\{0} we have that, .
Then we have from Lemma 6.1, This shows (6.3).
Assume F F ∈ Rid N . Then let For α ∈ N write P α (x) = p α x p α for the isotypical cut-down. For α, β, γ ∈ N the fusion rules of O + N (F) imply that if γ ≤ |α − β| and γ − α + β ∈ 2Z then γ is contained in α ⊗ β with multiplicity 1. We shall write V α,β γ : H γ → H α ⊗ H β for the isometry that intertwines γ with α ⊗ β. By Peter-Weyl theory V α,β γ is uniquely determined up to a complex scalar of modulus 1. For the next lemma let u α i, j denote the matrix unit of u α with respect to some orthogonal basis vectors which we simply denote by 1 ≤ i, j ≤ n α . We have Peter-Weyl orthogonality relations for some positive matrix Q α ∈ M n α (C) which may assumed to be diagonal after possibly changing the basis (see [28,Proposition 2.1]). Moreover we have, Lemma 6.5 Assume F F ∈ Rid N . Take matrix coefficients x = u α i, j , a = u r m ,n , c = u s m,n where α, r , s ∈ N. Assume r , s ≤ α and let k, l ∈ Z be such that |k| ≤ r and |l| ≤ s wit k − r ∈ 2Z and l − s ∈ 2Z. We have, P α+k+l (P α+k (cx)a) − P α+k+l (cP α+l (xa)) 2 q α x 2 . (6.4) Here is an inequality that holds up to a constant only depending on a, c and q.
Proof We prove this by induction on s and r . If either s = 0 or r = 0 the statement is clear as the left hand side of (6.4) is 0 Step 1. Case r = 1 and s = 1. We get the following. We have, Similarly, In fact by [64, Lemma A.2, Eqn. (A.5)] the left hand side of (6.7) may also be estimated by q α+(l−r )/2 in case l ∈ {−1, 1} and k = 1. Therefore for any k, l ∈ {−1, 1} except for k = l = −1 we may continu as follows. We get, Similarly, Combining all the above estimates yields, still with k, l ∈ {−1, 1} but not k = l = −1, P α+k+l (P α+k (cx)a) − P α+k+l (cP α+l (xa)) 2 ≤ 2q α x 2 a 2 c 2 . (6.10) But as we have that We can estimate the complementary case k = l = −1 through (6.10) by This proves the lemma in case s = r = 1.
Fix l ∈ Z with |l| ≤ s and l − s ∈ 2Z. Taking the sum over all l 1 and l 2 with l 1 +l 2 = l we see P α+k+l (P α+k (cx)a 1 a 2 ) − P α+k+l (cP α+l (xa 1 a 2 )) 2 q α x 2 . (6.11) Since a 1 and a 2 were arbitrary coefficients of u s−1 and u 1 respectively we get that (6.11) holds with a 1 a 2 replaced by any a that is a matrix coefficient of u (s−1)⊗1 . Since we have an inclusion of irreducible representations s ⊆ (s − 1) ⊗ 1 we conclude our claim.
Step 3. Case r and s arbitrary as in the lemma. One may proceed as in Step 2 to conclude the proof. Alternatively, assume the lemma is proved for r − 1 and s. We want to show that it holds for r and s. We have, Recall that every element in Irr(O + N (F)) is equivalent to its contragredient representation. So by the inductive step in Step 2 of the proof with the roles of s and r interchanged we see that the right hand side may be estimated by a constant only depending on a, c and q times q |α| .
The next lemma is now crucial. The fact that in the non-tracial case the Hilbert-Schmidt properties of the maps t in this lemma are better comes from the fact that the intertwining properties of Lemma 6.5 are stronger.
For t ≥ 0 consider the map If t > 0 then (l,2) t extends to a Hilbert-Schmidt map. Moreover, if F = Id N then (l,2) t extends to a Hilbert-Schmidt map also for t = 0.
Proof Let a and b in A be coefficients of respectively irreducible representations u r and u s with r , s ∈ N. By linearity it suffices to show that for t > 0 (and t = 0 if F = Id N ) the map, Note that each isotypical projection P γ , γ ∈ N commutes with which we may naturally view as a map A(O + N (F)) → A(O + N (F)). From the fusion rules of O + N (F) we conclude the following for numbers γ ∈ N. If α + γ ⊆ α ⊗ r then |γ | ≤ r . If α + γ ⊆ s ⊗ α then |γ | ≤ s. For |γ | ≤ r and β ⊆ s ⊗ (α + γ ) we have |β − α| ≤ r + s. Finally for |γ | ≤ s and β ⊆ (α − γ ) ⊗ r we have |β − α| ≤ r + s. These observations show that we get the following sum decomposition. Some summands can be zero; in fact all that matters is that the summation is finite. So, We therefore obtain for t > 0 that, We write for an inequality that holds up to some constant independent of α. Let γ, α, β be such that |β − α| ≤ r + s and |γ | ≤ max(r , s). Lemma 6.4 shows that, As the eigenvalues of grow asymptotically linear, more precisely Lemma 6.1, we have the following.
By Lemma 6.5 (note that b * is a coefficient of the contragredient of u s which is equivalent to u s itself), Combining (6.12) with the estimates from (6.13), (6.14) and (6.15) we see that, Now let ξ ∈ ⊕ N α=0 P α (L ∞ (G)) and let ξ α = P α (ξ ). Then Hence 0 is bounded L 2 (M) → L 2 (M). Further we get that, As n 2 α α q 2 converges to a number ≤ 1 (see Sect. 2) for α → ∞ this summation is finite as soon as t > 0 which concludes the proof. Moreover, if F = Id N then n 2 α α q 2 converges to a number < 1 (see Sect. 2) which concludes that the latter summation is finite if t = 0.
As a direct consequence we get the following.
. We first obtain the following result, which is closely related to Property(HH) + from [48] and its quantum version which was first studied in [31]. In fact Corollary 7.2 was already proved in [31,Corollary 4.7] based on different methods.
2 ∂ 2 has compact resolvent. With slight abuse of notation we will write ∂ for ∂ 2 as was also done in [48] and [50].  Lemma 3.9 shows that this derivation is closable with suitable domain so that = ∂ * ∂. Then Lemma 6.1 shows that ∂ is proper.

Corollary 7.2 Assume that F = id N . There exists a proper closable derivation ∂ on
The following Corollary 7.3 follows by a modification of the arguments in [48] from groups to quantum groups. This fact was also suggested in the final remarks of [31]. For completeness and the fact that in the non-tracial case we also require this result (even for stable normalizers), we included the proof in the appendix.

Strong solidity for O + N (F) and U + N (F), case of general F
Recall that for a matrix F ∈ G L n (C) the free unitary quantum group U + N (F) is defined as follows. As a C * -algebra it is the algebra A freely generated by elements u i, j , 1 ≤ i, j ≤ N subject to the relation that the matrix u 1 = (u i, j ) i, j is unitary and

The comultiplication is then given by
where z denotes the identity function on T = Z. Further, Wang [68] proved the following decomposition results. For any F ∈ G L N (C) we have an isomorphism of quantum groups and for certain matrices D i and E i of dimension N i and M i smaller than N respectively with the property that D i D i ∈ Rid N i and E i E i ∈ Rid M i .

Remark 7.4
Recall that in Proposition 6.2 we constructed a Markov semigroup ( t ) t≥0 on O + N (F) in case F F ∈ RId N . Then taking the free product with the identity semigroup on L ∞ ( Z) yields a semi-group on Z * O + N (F) which restricts to U + N (F) under the embedding (7.1). The gradient module of the identity semi-group is the zero module which is clearly IGHS. Therefore by Proposition 5.1 the free product semigroup is IGHS on Z * O + N (F) and hence on U + N (F). Take the free product of the latter semigroup on the U + N -factors in (7.2) and (7.3) and of the semigroup ( t ) t≥0 on the O + N -factors. This yields a semi-group of central multipliers on an aribtrary quantum group U + N (F) or O + N (F) that is moreover IGHS.
In the following proposition we collect some results from [40,49,64] that were not stated explicitly. We refer to [40] and [64] for the definition of bi-exactness and the Akemann-Ostrand property which shall not be used further in this paper. Proof By [24,Theorem 24] the reduced C * -algebras C r (O + N (F)) and C r (U + N (F)) have the CMAP and hence so do their free products [34,56]. This shows that such C * -algebras arelocally reflexive by [13,55,Chapter 18]. By [40, Theorem C] the (separable) quantum groups O + N (F), U + N (F) and their free products are bi-exact so that by [64,Theorem 2.5] (see also [49]) they are solid.
The following proposition is essentially [7,Main Theorem]. Let Z (M) = M∩M denote the center of a von Neumann algebra. Suppose that Q is a von Neumann subalgebra of M. Then we set the stable normalizer, For two faithful normal states ϕ and ψ on M we set to be the * -homomorphism given by π ϕ,ψ (u s ) = u s and π ϕ,ψ (π ψ (x)) = π ϕ (x) where s ∈ R, x ∈ M.
t = e −t ) t≥0 with ≥ 0 is called immediately L 2 -compact if the generator has compact resolvent.

Proposition 7.9 Let G be a compact quantum group and let M = L ∞ (G). Suppose that M is solid with the CMAP. Suppose moreover that M posseses a Markov semigroup of central multipliers that is both IGHS and immediately L 2 -compact. Then M is strongly solid.
Proof We follow the proof of [7,Main Theorem]. Let Q ⊆ M be a diffuse amenable von Neuman subalgebra with expectation. We need to prove that P = N M (Q) is amenable. Fix a faithful state ψ on M such that Q is globally invariant under σ ψ . The second paragraph of the proof of [7,Main Theorem] shows that by solidity of M we may replace Q by the amenable ψ-expected von Neumann subalgebra Q = Q (Q ∩M) and prove that N M ( Q) is amenable. This shows that without loss of generality we can assume that Q ∩ M = Z (Q). From this property it follows that c ψ (P) ⊆ N c ψ (M) (c ψ (Q)) , see [7,Section 4,Claim], and this inclusion is ψ-expected where ψ was the dual weight of ψ. Hence we need to prove that N c ψ (M) (c ψ (Q)) is amenable. Set P 0 = π ϕ,ψ (N c ψ (M) (c ψ (Q)) ), Q 0 = π ϕ,ψ (c ψ (Q)) and M 0 = c ϕ (M) = π ϕ,ψ (c ψ (M)). We have P 0 = N M 0 (Q 0 ) .
To show that P 0 is amenable it suffices to show that for every τ -finite projection p ∈ L ϕ (R) the von Neumann algebra pP 0 p is amenable. Let p ∈ L ϕ (R) be such a τ -finite projection. pP 0 p is contained with expectation in sN pM 0 p ( pQ 0 p) . So we need to show that sN pM 0 p ( pQ 0 p) is amenable.
Note that pQ 0 p isamenable [1]. Further, By [39, Lemma 2.5] we see that as Q is diffuse and p is τ -finite, we have pQ 0 p ⊀ pM 0 p L ϕ (R) p.
As M is equipped with a Markov semi-group of central multipliers that is IGHS, it follows that pM 0 p carries a GC semi-group, see Proposition 5.4. Moreover, by the same Proposition 5.4 and the discussion at the end of Sect. 2 (see [21]) we see that on pM 0 p there exists a closable derivation ∂ into a pM 0 p-pM 0 p bimodule that is weakly contained in its coarse bimodule of pM 0 p. Moreover the derivation is real (Lemma 3.10) and satisfies ∂ * ∂ = where is the generator of the GC semi-group constructed in Lemma 5.4, which on ( pL 2 (R)) ⊗ L 2 (M) is given by p ⊗ with the generator of the IGHS semi-group on M. By assumption has compact resolvent so that ∂ as a derivation on pM 0 p satisfies the properness assumption (A.1) with L = pL ϕ (R). As further M hence pM 0 p has the CMAP, we may apply Theorem A.5 to the triple ( pc ϕ (M) p, pL ϕ (R) p, pQ 0 p) to conclude that sN pM 0 p ( pQ 0 p) is amenable.

Remark 7.11
Anywhere in this paper the usage of semi-groups of central multipliers can be replaced by more general semi-groups of modular multipliers, i.e. multipliers : L ∞ (G) → L ∞ (G) that commute with the modular group σ ϕ of the Haar state. In turn one may apply averaging techniques to assure the existence of such semi-groups associated with quantum groups, c.f. [18,Proposition 4.2].
The reason that we must work with multipliers in this paper is to assure that the gradient bimodules H ∂ extend from A-A-bimodules to normal L ∞ (G)-L ∞ (G)-bimodules. It would be nice to have a more conceptual understanding in the general context of von Neumann algebras for when this happens.

Related results: amenability and equivariant compressions
We collect some final corollaries. Firstly, we recall the following result from [22]. We give their proof in terms of Stinespring dilations. then M is amenable.
Proof As Q is a conservative completely Dirichlet form there exists a Markov semigroup ( t ) t≥0 on M such that (2) t = e −t Q . (8.1) implies that for any K > 0 we find for large n that e λ n > n K . So if K > t −1 we see that for large n we get e −tλ n < n −1 . So e −t Q is Hilbert-Schmidt. Let (H t , η t ) be the pointed Stinespring M-M-bimodule of t . By Lemma 2.3 for every t > 0 we have H t is weakly contained in the coarse bimodule of M. As t → t is strongly continuous we get that H 0 is weakly contained in the coarse bimodule. Then as 0 = Id M , H 0 is the identity bimodule and so M is amenable.
As explained in the discussion before [7,Proposition 3.6], is that we have an equality sN • M (Q) = sN M (Q) . We shall call the latter von Neumann algebra P. Moreover, the partial isometries in sN • M (Q) generate P. Now we need the following weak compactness type property obtained in [7, Proposition 3.6].
Proposition A.1 Let (M, τ ) be a finite von Neumann algebra with the CMAP. Let Q be an amenable von Neumann subalgebra of M. Then there exists a net of positive vectors η n ∈ L 2 (Q ⊗ Q op ) such that (1) lim n (a ⊗ 1)η n − (1 ⊗ a op )η n 2 = 0, for all a ∈ Q.

A.2 Derivations
Now suppose that ∂ is a closable derivation on some σ -weakly dense * -subalgebra ⊆ M into a M-M-bimodule H. Moreover, assume that ∂ is real. Let ∂ be its closure. By [27,57] (so in the tracial case) we have that Dom(∂) ∩ M is still a σ -weakly dense * -subalgebra on which ∂ satisfies the Leibniz rule. Replacing ∂ by ∂ we may assume without loss of generality that ∂ is closed. We introduce notation (see [48,50]), for α > 0, Let e L be the Jones projection of M onto L; it is the map x τ → E L (x) τ with E L : M → L the τ -preserving conditional expectation. We further assume the type of properness assumption: Since ζ α i is ucp we see that (A.6) is larger than (A.8) which concludes the clam.