Smooth Dense Subalgebras and Fourier Multipliers on Compact Quantum Groups

We define and study dense Frechet subalgebras of compact quantum groups realised as smooth domains associated with a Dirac type operator with compact resolvent. Further, we construct spectral triples on compact matrix quantum groups in terms of Clebsch–Gordon coefficients and the eigenvalues of the Dirac operator D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{D}}$$\end{document}. Grotendieck’s theory of topological tensor products immediately yields a Schwartz kernel theorem for linear operators on compact quantum groups and allows us to introduce a natural class of pseudo-differential operators on them. It is also shown that regular pseudo-differential operators are closed under compositions. As a by-product, we develop elements of the distribution theory and corresponding Fourier analysis. We give applications of our construction to obtain sufficient conditions for Lp − Lq boundedness of coinvariant linear operators. We provide necessary and sufficient conditions for algebraic differential calculi on Hopf subalgebras of compact quantum groups to extend to our proposed smooth subalgebra CD∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{C}^\infty_\mathcal {D}}$$\end{document}. We check explicitly that these conditions hold true on the quantum SU2q for both its 3-dimensional and 4-dimensional calculi.


Introduction
In [HL36] Hardy and Littlewood proved the following generalisation of the Plancherel's identity on the circle T, namely (1.1) Hewitt and Ross [HR74] generalised this to the setting of compact abelian groups. Recently, the inequality has been extended [ANR15a] to compact homogeneous manifolds. In particular, on a compact Lie group G of topological dimension n, the result can be written as where π are obtained from eigenvalues of the Laplace operator G on G by I − G π i j = π π i j , i, j = 1, . . . , d π .
In [You08] the Hardy-Littlewood inequality has been extended to compact matrix quantum groups of Kac type. For this purpose, the author introduced a natural length function and extended the notion of 'rapid decay' to compact matrix quantum groups of Kac type. On the other hand, the inequality (1.2) on a compact Lie group G can be given a differential formulation as where G is the Laplacian on G and F G is the group Fourier transform. In the view of [Lic90], the operator I − G is the square of the spinor Dirac operator restricted to smooth functions on G. Thus, one can view the identity as associated to a 'Dirac-like' operator understood broadly. The Dirac operator was first introduced in 1928 by Paul Dirac to describe the evolution of fermions and bosons and plays an essential role in mathematical physics and representation theory. The geometric Dirac operator D can be constructed on an arbitrary spin Riemannian manifold (M, g) and Alain Connes showed [Con13] that most of the geometry of (M, g) can be reconstructed from the Dirac operator characterised abstractly as a 'spectral triple' (C ∞ (M), D, L 2 spin (M)). The axioms of a spectral triple in Connes' sense come from KO-homology. However, it is known that q-deformed quantum groups and homogeneous spaces do not fit into Connes axiomatic framework [Con95] if one wants to have the correct classical limit and various authors have considered modification of the axioms. Another problem is that of 'geometric realisation' where a given spectral triple operator (understood broadly) should ideally have an interpretation as built from a spin connection and Clifford action on a spinor bundle. A unified algebrogeometric approach to this has been proposed in [BM15], starting with a differential algebra structure on a possibly noncommutative 'coordinate algebra' and building up the geometry layer by layer so as to arrive at a noncommutative-geometrically realised D as an endpoint.
Compact quantum groups are quantisations of Poisson Lie groups, and it is natural to expect that every compact quantum group should possess a spectral triple (A, H, D) by a quantisation process of some sort. Following an approach suggested by [CL01], Chakraborty and Pal constructed [CP08] a spectral triple on the quantum SU q 2 . A Dirac operator agreeing with a real structure on SU q 2 has been suggested in [DLS + 05], which required a slight modification on the spectral triple axioms. More recently, inspired in part by [Fio98], Nesheveyev and Tuset constructed [NT10] spectral triples on the q-deformation G q of a compact simply connected Lie group G. To the best of our knowledge, it seems to be an open question whether there exists a spectral triple on an arbitrary compact quantum groups. And there remains the question of linking proposed spectral triples to the geometric picture. At the root of this problem is how to marry the analytic considerations of compact quantum groups to the differential-algebraic notion of differential calculus in the more constructive approach.
An outline of the results is as follows. After the set-up in Sect. 2 of quantum group Fourier transform, Paley-type inequalities in compact quantum groups G of Kac type are developed in Sect. 3 following the classical case in [ANR15a]. Section 4 studies left Fourier multipliers A (i.e. translation coinvariant operators) with associated symbol σ A . In Sect. 5 we introduce a 'Dirac operator' D : L 2 (G) → L 2 (G) defined by a sequence {λ π } of eigenvalues according to the Peter-Weyl decomposition of G. In Theorem 5.1 we obtain a formal version of Hardy-Littlewood inequaity (1.2) by applying Paley inequality (3.2) and write this in a differential form formulation where an unbounded D : L 2 (G) → L 2 (G) is defined by Dπ = λ π π and |D| −β is trace class.
This goes some way towards (G, L 2 (G), D) as a spectral triple in the sense of Alain Connes, this being the case if D has bounded commutators. We analyse in Theorem 5.3 when D defined by {λ π } is an actual spectral triple in the case of compact matrix quantum group in the sense of Woronowicz [Wor87]. Theorem 5.3 is then illustrated on the example of [CP08] on SU q 2 with eigenvalues ±(2l + 1) in the spin l part of the decomposition.
Section 6 studies smooth domains C ∞ D = α>0 Dom(|D| α ) associated with Dirac operator D on compact quantum groups. We study elements of distributions and rapid decay using C ∞ D (G). Grotendieck's theory of topological tensor products immediately yields a Schwartz kernel theorem for linear operators on C ∞ D (G) and allows us to introduce a natural class of pseudo-differential operators in this context. In Theorem 6.13 the Schwartz kernel K AB of composition of two regular pseudo-differential operators A, B : C ∞ D → C ∞ D is computed in terms of the kernels K A and K B . In Theorem 6.14 we show that regular pseudo-differential operators acting via right-convolution kernel can be represented by their global symbols (see Definition 6.11). A similar construction of smooth domains for general operators in Hilbert spaces has been carried out in [RT16].
Section 7 looks at how this notion of C ∞ D relates to the algebraic notion of differential 1-forms in the algebraic side of noncommutative differential geometry. We show that both the standard 3D left covariant and 4D bi-covariant differential calculi on C[SU is a q-integer. Thus our approach to 'smooth functions' is compatible with these qdifferential calculi, marrying the analytic and algebraic approaches. This q-deformed choice of D no longer obeys the bounded commutators condition for a spectral triple but is a natural q-deformation of our previous choice. On the other hand it is more closely related to the natural q-geometrically-realised Dirac [Maj03] and square root of a Laplace [Maj15] operators on SU q 2 which similarly have eigenvalues modified via q-integers.
The authors wish to thank Yulia Kuznetsova for her advice and comments. We also want to express our gratitude to the anonymous referees for their helpful suggestions.

Preliminaries
The notion of compact quantum groups has been introduced by Woronowicz in [Wor87]. Here we adopt the defintion from [Wor98].
Definition 2.1. A compact quantum group is a pair (G, ) where G is a unital C *algebra, : G → G ⊗ G is a unital, -homomorphic map which is coassociative, i.e.
The map is called the coproduct of G and it induces the convolution on the predual L 1 (G), Definition 2.2. Let (G, ) be a compact quantum group. A finite-dimensional representation π of (G, ) is a matrix [π i j ] in M n (G) for some n such that for all i, j = 1, . . . , n. We denote by G the set of all finite-dimensional irreducible unitary representations of (G, ).
Here we denote by M n (G) the set of n-dimensional matrices with entries in G. Let C[G] denote the Hopf subalgebra space of G spanned by the matrix elements π i j of finite-dimensional unitary representations π of (G, ). It can be shown [MVD98, Proposition 7.1, Theorem 7.6] that C[G] is a Hopf * -algebra dense in G. Every element f ∈ C[G] can be expanded in a finite sum where I f is a finite index set. It is sufficient to define the Hopf * -algebra structure on C[G] on generators π i j as follows is the antipode. These operations satisfy the usual compatiblity conditions with coproduct and product m G .
Every compact quantum group possesses [DK94] a functional h on G called the Haar state such that For every π ∈ G there exists a positive invertible matrix Q π ∈ C n π ×n π which is a unique intertwiner in Hom(π, π * ) such that Tr Q π = Tr (Q π ) −1 > 0. (2.2) We can always diagonalize matrix Q π and therefore we shall write Q π = diag(q π 1 , . . . , q π n π ).
It follows from (2.2) that which defines the quantum dimension d π of π . If G is a compact quantum group of Kac type, then d π = n π . The Peter-Weyl orthogonality relations are as follows (2. 3) The quantum Fourier transform F G : where L 1 (G) and L ∞ ( G) are defined below. The inverse Fourier transform F −1 G is given by From this it follows that { d π q π i π i j : π ∈ G, 1 ≤ i, j ≤ n π } is an orthonormal basis in L 2 (G). The Plancherel identity takes the form (2.5) We denote by C(π ) the coefficients subcoalgebra C(π ) = span{π i j } n π i, j=1 . The Peter-Weyl decomposition on the Hopf algebra C[G] is of the form (2.6) Let L 2 (G) be the GNS-Hilbert space associated with the Haar weight h. We denote by L ∞ (G) the universal von Neumann enveloping algebra of G. The coproduct and the Haar weight h can be uniquely extended to L ∞ (G). In general, there are two approaches to locally compact quantum groups: C * -algebraic and von Neumann-algebraic. Let ψ be a normal semi-finite weight on the commutant [L ∞ (G)] of the von Neumann algebra L ∞ (G). Let L 1 (G, ψ) be the set of all closed, densely defined operators A with polar decomposition A = u |A| such that there exists positive φ ∈ L ∞ (G) * and its spatial derivative dφ dψ = |A|. Setting A L 1 (G) = φ(1) = φ L ∞ (G) * yields an isometric isomorphism between L 1 (G, ψ) and L ∞ (G) * . Analogously, we denote by L p (G, ψ) the set of all closed, densely defined operators x such that there exists φ ∈ L 1 (G, ψ) such that |A| p = dφ dψ with the L p -norm given by A L p (G) = φ(1) 1 p . These spaces are isometrically isomorphic to the Haagerup L p -spaces [Haa79] and are thus independent of the choice of ψ.
One can introduce the Lebesgue space p ( G) on the dual G as follows Definition 2.3. We shall denote by p ( G) the space of sequences {σ (π)} π ∈ G endowed with the norm Here by the Hilbert-Schmidt norm we mean It can be shown that these are interpolation spaces in analogy to a similar family of spaces on the unitary dual of compact topological groups. The latter spaces were introduced in [RT10]. In this notation we can rewrite (2.5) as (2.11) Using the Hilbert-Schmidt norm and unitarity ππ * = Id of π 's, one can show (2.12) Hence, by the interpolation theorem for 1 < p ≤ 2 we obtain two versions of Hausdorff- (2.14) A Hausdorff-Young inequality on locally compact quantum groups has been obtained in [Coo10] and its sharpness explored in [LWW17]. We also present a version of a Marcinkiewicz interpolation theorem for linear mappings between compact quantum group G of Kac type and the space of matrix-valued sequences that will be realised via Thus, a linear mapping A : L ∞ (G) → takes a function to a matrix valued sequence, i.e.
We say that a linear operator A is of strong type ( p, q), if for every f ∈ L p (G), we have A f ∈ q ( G, ) and where M is independent of f , and the space q ( G, ) defined by the norm as in (2.7). The least M for which this is satisfied is taken to be the strong ( p, q)-norm of the operator A.
Denote the distribution functions of f and σ by μ G (t; f ) and ν G (y; σ ), respectively, i.e.
is said to be of weak type ( p, q); the least value of M in (2.18) is called weak ( p, q) norm of A. Every operation of strong type ( p, q) is also of weak type ( p, q), since The classical Marcinkiewicz interpolation theorem [Fol99, Theorem 6.28] has a natural analogue [ANR16, Theorem 6.1] for compact Lie groups. As a special case of [BC12, Theorem 2.1] we immediately obtain Theorem 2.4. Let 1 ≤ p 1 < p < p 2 < ∞. Suppose that a linear operator A from Ł ∞ (G) to is simultaneously of weak types ( p 1 , p 1 ) and ( p 2 , p 2 ), with norms M 1 and M 2 , respectively, i.e. (2.20) Then for any p ∈ ( p 1 , p 2 ) the operator A is of strong type ( p, p) and we have A different special case of [BC12, Theorem 2.1] has been presented in [You08].

Hausdorff-Young-Paley Inequalities
A Paley-type inequality for the group Fourier transform on commutative compact quantum group G = C(G) has been obtained in [ANR15a]. Here we give an analogue of this inequality on an arbitrary compact quantum group G of Kac type.
Theorem 3.1 (Paley-type inequality). Let 1 < p ≤ 2 and let G be a compact quantum group of Kac type. If ϕ(π) is a positive sequence over G such that The main tool in the proof of Theorem 3.1 is a noncommutative version of the Marcinkiewicz interpolation Theorem 2.4). Following the classical case in [ANR16] and [ANR15a], Youn obtained [You08] a different version of Paley type inequalities on compact quantum groups of Kac type. Theorem 3.2 (Hausdorff-Young-Paley inequality). Let 1 < p ≤ b ≤ p < ∞ and let G be a compact quantum group of Kac type. If a positive sequence ϕ(π), π ∈ G, satisfies the condition M ϕ := sup y>0 y π ∈ G ϕ(π)≥y n 2 π < ∞, (3.3) , and write L p (ω) = L p (ωdμ) for the weight ω. Suppose that 0 < p 0 , p 1 < ∞. Then where 0 < θ < 1, 1 p = 1−θ p 0 + θ p 1 , and ω = w From this, interpolating between the Paley-type inequality (3.2) in Theorem 3.1 and Hausdorff-Young inequality (2.13), we obtain Theorem 3.2. Hence, we concentrate on proving Theorem 3.1. The proof of Theorem 3.1 is an adaptation of the techniques used in [ANR15a].
We define the space L p (G, μ), 1 ≤ p < ∞, as the space of complex (or real) sequences a = {a l } l∈ G such that We will show that the sub-linear operator is well-defined and bounded from L p (G) to L p ( G, μ) for 1 < p ≤ 2. In other words, we claim that we have the estimate which would give (3.2) and where we set K ϕ := sup y>0 y π ∈G ϕ(π)≥y n 2 π . We will show that A is of weak type (2,2) and of weak-type (1,1). More precisely, with the distribution function Then (3.2) will follow by the Marcinkiewicz interpolation Theorem 2.4). Now, to show (3.6), using Plancherel's identity (2.10), we get Thus, T is of type (2,2) with norm M 2 ≤ 1. Further, we show that T is of weak-type (1,1) with norm M 1 = C; more precisely, we show that The left-hand side here is the weighted sum ϕ 2 (π )n 2 π taken over those π ∈ G for which f (π ) HS √ n π ϕ(π) > y.
From definition of the Fourier transform it follows that Therefore, we have Using this, we get Setting (3.9) We claim that (3.10) In fact, we have We can interchange sum and integration to get Further, we make a substitution τ = y 2 , yielding This proves (3.10). We have just proved inequalities (3.6), (3.7). Then by using Marcinkiewicz' interpolation theorem (see Theorem 2.4) with p = 1, q = 2 and 1 This completes the proof.

Fourier Multipliers on Compact Quantum Groups Definition 4.1. Let (G, ) be a compact quantum group. A linear operator
For a compact topological group this means an operator that commutes with left translation on the group and defines a global symbol σ A of A.
Theorem 4.2. Let G be a compact quantum group and let A : where σ A (π ) ∈ C n π ×n π are defined by Aπ = πσ A (π ).
Proof of Theorem 4.2. By the Peter-Weyl decomposition (2.6), it is sufficient to establish (4.2) on the coefficient sub-colagebra C(π ). Suppose A is left invariant and write for some coefficients c π i jkl . Then by left invariance, Comparing these, by the Peter-Weyl decomposition, we see that only π = π can contribute. Moreover, since the {π i j } are a basis of C(π ) we must have c π i jkl = 0 unless k = i. So we have l,m c π i jil π im ⊗ π ml = m π im ⊗ Aπ m j . Comparing these we see that and we check that which is the same. This proves Theorem 4.3.
Essentially similar arguments can be found in an earlier paper by [CFK14]. Also note from the proof that the same result applies to any coinvariant linear map A : . We refer to the operators A acting in this way on the Fourier side as quantum Fourier multipliers. In greater generality, such result was also shown in [JNR09] for every locally compact quantum group; the authors thank the referee for pointing out this reference. In the classical situation on G = T n , left Fourier multipliers are essentially operators acting via convolution with measures whose Fourier coefficients are bounded.
Let A : G → G be a left Fourier multiplier. We are concerned with the question of what assumptions on the symbol σ A guarantee that A is bounded from L p (G) to L q (G).

left Fourier multiplier. Then we have
(4.3)

Proof of Theorem 4.3. By definition
(4.5) The case q ≤ ( p ) can be reduced to the case p ≤ q as follows. The L p -duality yields and its operator norm σ A * (π ) op equals to σ A (π ) op . Now, we are in a position to apply Theorem 3.2. Set 1 r = 1 p − 1 q . We observe that with σ (π) := σ A (π ) r op I d π , π ∈ G, and b = q , the assumptions of Theorem 3.2 are satisfied and we obtain Further, it can be easily checked that where in the last equality we used the continuity of the s 1 r .

Hardy-Littlewood Inequality and Spectral Triples
As a corollary of Theorem 3.1, we obtain a formal compact quantum group version of the Hardy-Littlewood inequality by using a suitable sequence {λ π }. This is formal in the sense that we do not study underlying inherent geometric data, but nevertheless by formulating this data in terms of an operator D our quantum Hardy-Littlewood type inequality (5.2) presents in a manner similar to the compact Lie group inequality (1.3).
Theorem 5.1. Let 1 < p ≤ 2 and let G be a compact quantum group of Kac type. Assume that a sequence {λ π } π ∈ G grows sufficiently fast, that is, (5.1) In terms of an unbounded D : L 2 (G) → L 2 (G) defined by Dπ = λ π π , (5.1) and (5.2) are respectively equivalent to |D| −β trace class and Proof of Theorem 5.1. By the construction Then we have Then by Theorem 3.1, we get (5.2). For the second part, we equivalently view {λ π } as defining an operator D and in the case of Kac type, summability with power β reduces to condition (5.1). Clearly, Using this and the right-hand side in inequality (5.2), we obtain (5.3).
Although the operator D is not immediately geometric, it can fit into Connes noncommutative geometric framework of a 'spectral triple'. The infimum of β ∈ R + such that |D| −β+ε is trace-class for every ε > 0 is called the spectral dimension [Con96].
Definition 5.2 is very minimal in the sense that we do not impose any conditions on reality and chirality operators and their interrelations with D as in [Con95]. It is certainly possible for D in Theorem 5.1 to obey the further boundedness condition for a spectral triple in the Kac type case, a topic to be explored further elsewhere.
For now, we turn to an explicit construction of spectral triples more relevant to qdeformed examples. If π k , π s ∈ G (with labels as shown) then the tensor product π k ⊗π s is a completely reducible finite-dimensional representation. The matrix elements of π k ⊗ π s are given by We shall define the coefficients C ksw i j prut as follows It then follows from (5.6) where I ks is a finite subset of N. These Clebsch-Gordan coefficients are important to write down the action of the commutator ∂(a) = [D, a] explicitly. In [CP08], these coefficients were computed for the quantum groups SU q 2l+1 , allowing the authors compute the action of the left multiplication operator on L 2 (SU q 2l+1 ) and leading in turn to growth restriction on the eigenvalues λ k . In order to consider this more generally, we take a slightly different approach and leave the Clebsch-Gordan coefficients in the bound as they depend on the compact quantum group.
It is convenient, however, to focus on the compact matrix quantum group case [Wor87,MVD98]. Here there is a matrix of generators u mn of C[G] corresponding to a defining unitary representation. We can expand them as It can be clearly seen from the Peter-Weyl decomposition that there are only finitely many non-zero α i jk mn for each generator u mn . These coefficients are closely related to the Clebsch-Gordan coefficients.
holds for all s, m, n, r and some constant C.
Example 5.4 (Equivariant spectral triples on the quantum SU q n ). Condition (5.9) imposes certain growth condition on consecutive differences λ k − λ s(k) of the eigenvalues λ k . For G = SU q n , it is possible to compute [CP08] the coefficients C ksm i j prut . In more detail, the authors showed that the C ksm i j prtt 's are essentially powers of q, i.e.
is bounded on u mn if and only if condition (5.9) is true. Let u mn ∈ C[G], π s ∈ G and take a = u mn and b = π s pr . The action of D on u mn is as follows In particular, for a = u mn ∈ C[G] and b = π s pr ∈ C[G], we get ∂(u mn )π s pr L 2 (G) ≤ C a π s pr L 2 (G) . (5.14) Then by the direct computation (5.15) Using that π s pr 2 L 2 (G) = q s r d π r and (5.15), we obtain (5.9). The converse is also true. Indeed, writing inequality (5.15) in the reverse order, we get (5.14). We do not consider β-summability in Theorem 5.3. It is known to hold for some β in the case of Example 5.6, see [CP08]. Also note that we cannot immediately apply Theorem 5.3 to G = SU q 2 with λ k given more naturally by q-integers as we will do later. Indeed, it has been already observed in [Con08] that standard quantum group examples do not very naturally fit into the classical theory of spectral triples, which may need to be replaced by a twisted version for example in the commutator (5.11). In this context it is also possiblel to demand a 'quantum summability' condition for some β > 0, i.e. with respect to a modified 'quantum' trace. This reduces correctly for classical compact Lie groups and compact quantum groups of Kac type, and also holds for G = SU q 2 if we use q-integers for the λ k but not if we use the classical values in Example 5.6.

Schwartz Kernels
Definition 6.1 (Smooth domain). Let G be a compact quantum group and let D : C[G] → C[G] be a linear map extended to L 2 (G) → L 2 (G) as a closed unbounded linear operator. Then the smooth domain C ∞ D ⊂ G of D is defined as follows The Frechet structure is given by the seminorms The powers |D| α are defined by the spectral theorem. It can be checked that C ∞ D is a locally convex topological vector space.
We show that every linear operator A : C ∞ D → C ∞ D continuous with respect to the Frechet topology can be associated with a distribution K A 'acting' on G × G. In other words, every linear continuous operator A : C ∞ D → C ∞ D possesses a Schwartz kernel K A . This allows us to define the global symbol of A in line with the pseudo-differential calculus on compact Lie groups [RT13,RT10]. The global symbols have been recently studied [LNJP16] on the quantum tori T n θ , n ∈ N, θ ∈ R.
Definition 6.2 (Rapidly decreasing functions on G). Denote by S( G) the space of matrixvalued sequences {σ (π)}, σ (π) ∈ C n π ×n π satisfying the conditions The space S( G) becomes a locally convex topological space if we endow it with the norms The construction of the topology on C ∞ D (G) readily implies that the quantum Fourier transform F G is a homeomorphism between C ∞ D and S( G).
Proposition 6.6. The space S (G) is complete, i.e. for every Cauchy sequence {u n } ⊂ S (G) the limit u = lim u n ∈ S (G) exists and belongs to S (G). If ϕ n converges to ϕ in C ∞ D , then lim n u n (ϕ n ) = u(ϕ).
Both of these are by standard methods and hold in the general situation [Tre67]. By transposing the inverse Fourier transform F −1 G : S( G) → C ∞ D (G), the Fourier transform F G extends uniquely to a mapping . In other words, for every distribution u ∈ S (G) its Fourier transform F G [u] is a distribution on S( G).
Proposition 6.8. Let (G, ) be a compact quantum group and let |D| −β be of trace class for some β > 0. Then the Frechet space C ∞ D is nuclear.
Proof of Proposition 6.8. It is sufficient to prove that S( G) is a nuclear Frechet space since F G is a homeomorphism. The former fact follows from [Tri78, Section 8.2.1].
The theory of topological vector spaces has been significantly developed [Gro55] by Alexander Grothendieck. It turns out that the property of being nuclear is crucial and these spaces are 'closest' to finite-dimensional spaces. The nuclearity is the necessary and sufficient condition for the existence of abstract Schwartz kernels.
Definition 6.9. A linear continuous operator A : is called a pseudodifferential operator.
From the abstract Schwartz kernel theorem [Tre67], we readily obtain Theorem 6.10. Let D be as in Proposition 6.8. Let A : C ∞ D → S (G) be a pseudodifferential operator. Then there is a distribution K A ∈ S (G) ⊗S (G) such that The structure theorem [Tre67, Theorem 45.1] applied to the topological tensor product S (G) ⊗S (G) immediately yields that the Schwartz kernel K A can be written in the form where ∞ n=1 s A n < +∞ and {x n }, {t n } ⊂ S (G) tend to 0 in S (G). This allows us to define global symbols σ A in line with the classical theory.
Definition 6.11. Let A : C ∞ D (G) → S (G) be a pseudo-differential operator. We define a global symbol σ A of A at π ∈ G as a matrix σ A (π ) = [σ A (π i j )] n π i j of distributions σ A (π i j ) ∈ S (G) acting by the formula Alternatively, we have (6.6) Definition 6.12. We say that a pseudo-differential operator A : It can be easily seen that this class of pseudo-differential operators is closed under composition. Explicit composition formula for the global symbols on quatum tori T n θ has been recently obtained in [LNJP16]. In Theorem 6.13 we derive the composition formula in terms of abstract Schwartz kernels.
Theorem 6.13. Let A : C ∞ D → C ∞ D and B : C ∞ D → C ∞ D be two regular pseudodifferential operators on G. Let K A and K B be the Schwartz kernels of A and B. Then the composition A • B : C ∞ D → C ∞ D is a regular pseudo-differential operator. Moreover, the Schwartz kernel K AB of the composition AB is given by Proof of Theorem 6.13. Our proof relies on the theory of topological vector spaces. We notice that By explicit calculations, we have The injective topology on S (G) yields that x A n tends to 0 uniformly over bounded subsets of C ∞ D , i.e.
lim n x A n (x B m ) = 0.
Thus, using this and passing to the limit in (6.10), we obtain (6.9). We refer to [Tre67, Part III] for further details and definitions.
We introduce the right-convolution Schwartz kernel R A by the formula where u A n is a convolution type vector-valued distribution acting by the formula and s A n are as in (6.4).
Theorem 6.14. Let (G, ) be a compact quantum group and let A : C ∞ D → C ∞ D be a regular pseudo-differential operator acting via right-convolution kernel, i.e.
where σ A (π ) is the global symbol of A defined by (6.5).
Proof of Theorem 6.14. Let f ∈ C[G]. Then we have We shall start by showing that (6.12) holds true for f ∈ C[G].
We have Therefore, we get (6.12) where we can replace I f by G.

Differential Calculi on Compact Quantum Groups
In this section we are going to ask how the above 'Fourier approach' to the analysis on compact quantum groups interplays with the theory of differential structures on Hopf algebras of compact quantum groups and how this extends to C ∞ D (G). Recall that differential structures in the literature have been defined at the polynomial level i.e. on C[G] as a Hopf * -algebra. For every choice of {λ π } π ∈Ĝ we have an operator defined by Dπ = λ π π and Thus for G = SU q 2 we have C[G] = C q [SU 2 ] as the usual dense Hopf * -subalgebra of SU q 2 with a 2 × 2 matrix of generators while C ∞ D (SU q 2 ) lies in between as something more akin to C ∞ (SU 2 ). Our goal in the present section is to show that elements of C ∞ D (G) are indeed smooth with respect to a suitable differential structure at least for SU q 2 and in outline for the general q-deformation case. We start by recalling the purely algebraic definition of first-order differential calculus over associative algebras and refer to [Maj16] for a thorough exposition. Let A be a unital algebra over a field k.
Definition 7.1. A first order differential calculus ( 1 , d) over A means (1) 1 is an A-bimodule.
(2) d : A → 1 is a linear map satisfying (3) The vector space 1 is spanned by elements of the form adb, In the * -algebra case, * extends uniquely to 1 in such a way that it commutes with d.
Example 7.2. Let A = C ∞ (R) and 1 = C ∞ (R).dx with left and right action given by multiplication in C ∞ (R) (so functions and dx commute). The exterior derivative is d f = ∂ f ∂ x dx as this is the classical calculus. There are many other interesting calculi even on the commutative algebra of functions in one variable, see [Maj16]. This case was first analysed in [Wor89] but here we continue with a modern algebraic exposition. Note that the last two requirements imply that L (adb) = a (1) b (1) ⊗ a (2) db (2) using the Sweedler notation a = a (1) ⊗ a (2) , and conversely if this formula gives a well-defined map then one can show that it makes the calculus left covariant. Hence this is a property of ( 1 , d) not additional data. We have a similar notion of right covariance and the calculus is called bicovariant if it is both left and right covariant. Let 1 = {ω ∈ 1 | L ω = 1 ⊗ ω} be the space of left-invariant 1-forms on a left-covariant calculus.
In this case we define the Maurer-Cartan form : A + → 1 by . This map is surjective by the spanning assumption above and is a right A-module map, since where 1 is a right module by ω b = (Sb (1) )ωb (2) . Hence 1 ∼ =A + /I for some right ideal I ⊂ A + . Conversely, given a right A-module 1 and a surjective right-module map : A + → 1 we can define a left covariant calculus by exterior derivative and bimodule relations da = a (1) π ε a (2) , ωa = a (1) (ω a (2) ), ∀a ∈ A, ω ∈ 1 .
Here π ε (a) = a − ε(a) projects A → A + and 1 = A. 1 is free as a left module. If the calculus is bicovariant then 1 also has a right coaction making 1 an object in the braided category M A A of crossed A-modules (also called Radford-Drinfeld-Yetter modules). Here A + for any Hopf algebra is also a right crossed module by (the right adjoint coaction) and : A + → 1 becomes a surjective morphism of right crossed-modules in the bicovariant case. Now let the calculus be bicovariant and 1 of finite dimension n as a vector space ( 1 finite-dimensional over A) and {e i } n i=1 a basis of 1 with { f i } a dual basis. Then the associated 'left-invariant vector fields' (which are not necessarily derivations) are given by π ε a (2) ), ∀a ∈ A and obey ∂ i = (id ⊗∂ i ) as in Definition 4.1 but on A and da = i (∂ i a)e i . The global symbols σ i : A → k defined by σ i = f i , π ε ( ) can be recovered from ∂ i as σ i (a) = ε∂ i a and can typically be realised as evaluation against some element x i of a dually paired 'enveloping algebra' Hopf algebra and in this context we will write σ i = σ x i = x i , . Similarly for each i, j let C i j (a) = a (1) f j , e i a (2) , for a ∈ A, be left-invariant operators encoding the bimodule commutation relations. They have no classical analogue (they would be the identity). Their symbols σ i m : A → k defined by σ i j (a) = f j , e i a = εC i j (a) can typically be given by evaluation against elements y i j of a dually paired Hopf algebra and this in this context we will write σ i j = σ y i j = y i j , . It is these global symbols which we extract from the algebraic structure of the calculus and need in what follows. Now let (G, ) be a compact quantum group with dense * -Hopf subalgebra A = C[G]. We suppose that we have a left covariant calculus on C[G] and remember from it the key information 1 and the operators ∂ i , C i j defining the exterior derivative and bimodule relations respectively. ( 1 , d) be a ndimensional left-covariant differential calculus over the dense Hopf * -algebra

Proposition 7.4. Let (G, ) be a compact quantum group and let
The extension is given by and where the last step is the matrix of the representation of a dually paired Hopf algebra defined by π when such x k exist. Therefore, it is sufficient to check that ∂ k : C ∞ D → C ∞ D are continuous linear maps with respect to the topology defined by seminorms (6.2). By [Tre67, Proposition 7.7, p.64], the linear maps ∂ k act continuously in C ∞ D (G) if and only if for every α > 0 there is β > 0 such that for every a ∈ C ∞ D (G). It is clear that condition (7.1) implies (7.3). Hence, we concentrate on necessity. Taking a = π i j ∈ G in (7.3), we get λ 2α π ∂ k (π i j )(π ) 2 HS ≤ λ 2β π π i j (π ) 2 HS = λ 2β π .
(7.4) From (7.4) dividing by |λ π | 2α , we get ∂ k (π i j )(π ) 2 HS ≤ |λ π | 2(β−α) . (7.5) From the algebraic version of Theorem 4.3 we have where we used The latter follows from the Peter-Weyl orthogonality relations (2.3). Hence, we get (7.6) Thus, estimate (7.4) reduces to We similarly need to extend the bimodule relations from C[G] to C ∞ D (G) and we do this in just the same way by where from the above and the algebraic form of Theorem 4.3 we have and where the last step is the matrix of the representation of a dually paired Hopf algebra defined by π when such y i j exist. As before we need these linear maps C i j : C[G] → C[G] to extend to C ∞ D which is another Hilbert-Schmidt condition on the symbols of the same type as for the ∂ i .
Conversely, given a differential calculus ( 1 , d) over the Hopf-subalgebra C[G] of G, we shall view (7.1) as a restriction on the {λ π } i.e. on a 'Dirac operator' D : L 2 (G) → L 2 (G) for it to agree with the differential calculus ( 1 , d) over C[G].
Definition 7.5. Let D : L 2 (G) → L 2 (G) be unbounded operator given by Dπ mn = λ π π mn for some collection {λ π } π ∈ G . We shall say that D is admissible with respect to a differential calculus ( 1 , d) on C[G] if and only if the condition (7.1) on {λ π } π ∈ G holds for some γ > 0.
Whether or not an admissible D exists depends on the quantum group and the calculus. We look at SU q 2 with its two main calculi of interest, the 3D and the 4D (both of these calculi are from [Wor89] but the 4D one generalises to other q-deformation quantum groups). As a first step, we will recall the computation of the global symbols for the vector fields on the classical SU 2 . We first briefly recall representation theory of SU where t l = (t l mn ) m,n∈I l and I l = {−l, −l + 1, . . . , +l − 1, +l}, l ∈ 1 2 N 0 . It is convenient for the inverse Fourier transform to introduce q-traces τ l (σ (l)) := i∈I l q 2i σ (l) ii .
From this the classical global symbols σ ∂ ± , σ ∂ 0 can be read off as the matrix entries of X ± , H in the representation t l . The corepresentation theory of SU q 2 is strikingly similar to its classical counterpart giving similar results. We compute the symbols for the action of X − , X + , q H 2 as elements of the quantum enveloping algebra U q (su 2 ) acting by the regular representation on C q [SU 2 ] and in the conventions of [Maj95].
Lemma 7.7. We have Proof of Lemma 7.7. Let q be real and for each l ∈ 1 2 N 0 , the quantum group U q (su2) has 2l + 1-dimensional unitary representation space V l = {|l m } +l m=−l detailed in the relevant conventions in [Maj95, Proposition 3.2.6, p.92] so that, for example, X + |l, m = [l − n] q [l + n + 1] q |l, m + 1 . By definition, the t l mn are the matrix elements of this representation, immediately giving σ X (t l ) mn = X, t l mn = t l (X ) mn for the symbol of any left-invariant operatorX (a) = a (1) X, a (2) . Thus we can read off the σ X ± , σ q H 2 as stated.
For the convenience of the reader we recall that the Q-matrix for G = SU q 2 is given [MMN + 91] by As a warm-up we look at the admissibility condition (7.1) of Proposition 7.4.
Lemma 7.8. Let b q = max(q, 1 q ). Then We write x ∼ = y if there are constants c 1 , c 2 = 0 such that Lemma 7.9. Let G = SU q 2 . Then we have Proof of Lemma 7.9. By (2.8) where we used the fact that +l m=−l q m ∼ = b l q . (7.12) Similarly, we get Finally, we compute (7.14) This completes the proof.
By the arguments as in the proof of Proposition 7.4 it follows that the associated left covariant operators to X ± and q H 2 − 1 extend to C ∞ D (SU q 2 ) where Dt l i j = ±[2l + 1] q t l i j (7.15) since the || || 2 H S condition similar to (7.1) holds, but for the symbols of these operators rather than for a choice of calculus.
7.1. 3D calculus on SU q 2 . We are now ready for the left-covariant 3D calculus on C q [SU 2 ] which we take with the defining 2-dimensional representation with t α β = {a, b, c, d} to give the standard matrix of generators with usual conventions where ba = qab etc. We let | | denote the known Z-grading on the algebra defined as the number of a, c minus the number of b, d in any monomial. The 3D calculus has generators e 0 , e ± with commutation relations (which implies the action in the vector space 1 with basis e 0 , e ± ). The exterior derivative is and can in fact be identified as evaluation against elements x k ∈ U q (su 2 ) in our case.
is generated by the action of and extends to C ∞ D (SU q 2 ) where D defined in (7.15) is admissible with γ = 2. The symbols are given by Proof of Proposition 7.10. The 3D calculus is constructed 'by hand' so we use the form σ k = ε∂ k and the known form of the partial derivatives (obtained by computing d on monomials via the Leibniz rule) and find elements x k ∈ U q (su 2 ) as stated that give these. One then finds the symbols σ x k (π ) i j = π(x i ) i j as where we used the fact that σ X (π )σ Y (π ) = σ XY (π ) since these are matrices for X, Y, XY in the representation π , and Lemma 7.7. Similarly, we establish where for the haar function h we estimate It is then straightforward to check that the condition (7.1) is satisfied for the symbols σ x ± (t l ), σ x 0 (t l ). Hence, the application of Proposition 7.4 shows that the vector fields x ± , x 0 are continuous. Now, we check condition (7.1) allowing us to extend x + , x − , x 0 continuously. We have (7.17) Similarly We similarly have commutation relations given for i, j = ± by if we number the indices by −l, . . . , l for the 2l + 1 dimensional representation t l . This gives commutation relations e i t l jk = t l jm σ i p (t l ) mk e p = t l jm e i δ mk q −2k = t l jk q −2k e i for i = ± (and q 2 in place of q if i = 0) which corresponds to a Z-grading of C q [SU 2 ] where t l jk has grade −2k. For the spin 1/2 representation it means a, c in the standard matrix generators t mn of the quantum group have grade 1 and b, d have grade -1 as expected. We compute σ i j (t l ) 2 HS similarly as in the proof of Lemma 7.9. By (7.21) (7.18) 7.2. 4D calculus on SU q 2 . As before we denote the standard 2 × 2 matrix of generators of C q [SU 2 ] by a, b, c, d. This time (from the general construction given later or from [Wor89]) there is a basis e a , e b , e c , e d corresponding to the generators, with relations and exterior derivative Here [x, y] q ≡ x y − qyx and λ = 1 − q −2 . Then the generators of the calculus are elements x k ∈ U q (su 2 ) where k = a, b, c, d which we organise as a 2 × 2 matrix of elements (x αβ ) where These combinations x αβ are known to span the right handed braided-Lie algebra L ⊂ U q (su 2 ) and generate the quantum group [Maj15].
Proof of Proposition 7.11. We compute the symbol σ x αβ (π ) of x αβ composing the results in Lemma 7.7 to find It is sufficient to check that x αβ acts continuously in each L 2 (G). Let us denote By (2.8) (7.21) Composing (7.21) and (7.20), we get = (q 4l + 2q −2 + q 4 l−4 + 2q 2l + 2q −2l−2 + 1)(2l + 1) The expression −2q 2l−2 − 2q −2l−4 − 2q −2 is always negative. Therefore, we get σ a (t l ) 2 HS ≤ q 4l + 2q −2 + q −4l−4 + 2q 2l + 2q −2l−2 + 1 (2l + 1) It is straightforward to check that where in the first inequality we used the fact In the second inequality we used Lemma 7.8 with n = 2l + 1. Now, we compute (7.26) We can argue analogously for σ c (t l ) HS to get Finally, one checks by direct calculation that (7.28) Now, we check that the matrices σ αβ γ δ encoding the bimodule commutation relations in the 4D calculus satisfy condition (7.1) with some exponent γ . The bimodule relations are best handled as part of a general construction discussed later and from (7.30) and (7.31) there, we see the seven values The non-zero matrices σ αβ γ δ are obtained by reading (7.31) and plugging it into (7.30), noting that S X − = −q −1 X − Sq H 2 = q − H 2 for the action of the antipode. We then compute the symbols by composing the symbols for the composition of invariant operators, to obtain Now we can compute the corresponding q-deformed Hilbert-Schmidt norms, The application of Proposition 7.4 completes the proof that d extends.
7.3. Generalising to other coquasitriangular Hopf algebras. The bicovariant 4D calculus on A = C q [SU 2 ] is an example of a canonical construction whenever A is coquasitriangular in the dual of the sense of V.G. Drinfeld, i.e. a map R : A ⊗ A → C obeying certain axioms. This gives a bicovariant calculus for any L ⊂ A a subcoalgebra [Maj15]. We define which we view as = Q : A + → 1 = L * . If this is not surjective we take 1 to be the image, but in examples it tends to be surjective so we suppose this as a property of the data (A, R, L). In addition L is canonically a left crossed A-module [Maj15] which makes 1 a right crossed A-module with a morphism. Here the left action on L is The simplest case of interest is when L is the span of the matrix elements of a corepresentation t ∈ A, L = span{t α β }. We let {e αβ } be the dual basis of 1 , so f αβ = t α β is the dual basis element to e αβ . We let e α be a basis of the corepresentation V so R e α = e β ⊗ t β α . The associated left representation of any Hopf algebra U dually paired to A is t (x) α β = t α β , x for all x ∈ U or x.e α = e β t β α , x . In this case if π i j are the matrix elements of a representation π ∈ A then σ αβ (π ) i j = f αβ ( (π i j − δ i j )) = Q(π i j ⊗ t α β ) − δ i j δ α β (7.29) which we can usually write as for some elements x αβ ∈ U for suitable U . Here x αβ = ((Sl − )l + ) α β − δ α β in the quantum groups literature [Maj95] for certain elements l ±α β ∈ U . These elements are evaluated in the associated matrix representation π of U and ε(x α β ) = 0 is implied by the above. Similarly, the adjoint of the action on L gives the right action e αβ a = e γ δ R(t γ α ⊗ a (1) )R(a (2) ⊗ t β δ ) = e γ δ a, Sl −γ α l +β δ .
Hence the action of matrix elements π i j of a corepresentation is e αβ π i j = e γ δ π((Sl −γ α )l +β δ ) i j , or in terms of the matrix that governs the commutation relations, this is σ αβ γ δ (π ) i j = π((Sl −γ α )l +β δ ) i j . (7.30) For the example of U q (su 2 ) one has [Maj95] giving the formulae for x α β previously used. It seems clear that this calculus will similarly extend to C ∞ D (G) for the general q-deformation of a compact simple group with A = C[G] coquaistriangular. Details will be considered elsewhere.

Concluding Remarks
Having a suitable summable D to define a smooth subspace C ∞ D to which the differential calculus extends, as above, is an important step towards an actual geometric Dirac operator. In the coquasitriangular case with the bicovariant calculus defined by a matrix corepresentation, we have 1 = End(V ) for some comodule V and following [Maj03] we can define 'spinor sections' S ∞ D = C ∞ D ⊗ V and a canonical map where {e α } is a basis of V and s α ∈ C ∞ D . At the algebraic level this was in [Maj03], but since the partial derivatives extend to 'smooth functions' we see that so does D to our 'smooth sections'. This was studied at the algebraic level in detail for A = C q [SU 2 ] and justified as a natural Dirac-like operator that bypasses the Clifford algebra in the usual construction of the geometric Dirac operator, and fits with that after we add an additional constant curvature term (a multiple of the identity). Using our results for this quantum group we have D αt l mn βt l pr = αt l ms σ 11 (t l sn ) + βt l ps σ 12 (t l sr ) αt l ms σ 21 (t l sn ) + βt l ps σ 22 (t l sr ) for coefficients α, β and for the symbols (7.20) given previously (we sum over s in the appropriate range). The eigenvalues of the geometrically normalised D/λ when restricted to both spinor components in the Peter-Weyl subspace spanned by {t l mn } are (i) q l+1 [l] q ; (ii)(l > 0) − q −l [l + 1] q and fully diagonalise this subspace of dimension 2(2l +1) 2 , and hence together fully diagonalise D. The type (i) eigenvalues were already noted for the reduced Hopf algebras at odd roots unity in [Maj03,Prop. 5.2] in the equivalent form q 2 [2l; q]/[2; q] = q 2 [l; q 2 ], where [m; q] = (q m −1)/(q −1) = 1+q +· · ·+q m−1 . We see using our Fourier methods that we also have a second set (ii) both at roots of unity (beyond the 3rd root) and for real or generic q. Note that our above geometric D is not directly comparable to our operators D because our spinor space is two-dimensional so that D does not act on one copy of the coordinate algebra, and nor should it geometrically, but is in the same ball park as the q-deformation (7.15) of the classical D with eigenvalues 2l + 1 discussed in Example 5.6. Also note that the bicovariant matrix block calculi are typically inner in the sense of a nonclassical direction θ such that [θ, f ] = λd f , and that is the case for the 4D calculus on C q [SU 2 ] with θ = e a + e d . One can choose a more geometric basis e z , e b , e c , θ where the first three have a classical limit as usual and e z = q −2 e a − e d . The partial derivative ∂ θ for the θ -direction in this basis turns out to be the q-deformed Laplacian q as explained in [Maj15]. There is a quantum metric g = e c ⊗ e b + q 2 e b ⊗ e c + q 2 q + q −1 (e z ⊗ e z − θ ⊗ θ) and denoting its coefficients as g i j one has a natural q-Laplace operator [Maj15] q = q 2 g i j ∂ i ∂ j , ∂ θ = q∂ a + q −1 ∂ d q + q −1 = q 2 λ 2 q + q −1 q (where we have changed to our more geometric normalisation of ∂ i and d). Once again, since we have seen that the partial derivatives extend to C ∞ D (SU q 2 ), this q also extends and, using our result (7.20), we are in a position to compute it in our Peter-Weyl basis as q t l mn = [2] q q 2 λ 2 ∂ θ t l mn = q −2 λ −2 (qt l ms σ 11 (t l sn ) + q −1 t l ms σ 22 (t l sn )) = q(q 2l − 1) + q −1 (q −2l − 1) (q − q −1 ) 2 t l mn = [l] q [l + 1] q t l mn for m, n = −l, . . . , l. One could then take a square root involving q much as in Example 5.5 for the operator D to provide the smoothness. Further q-harmonic analysis using our Fourier methods will be considered elsewhere to include smooth functions and harmonic analysis on the q-sphere obtained from the 3D differential calculus on C ∞ D (SU q 2 ), extending the algebraic line for the geometric Dirac operator on the q-sphere in [BM15]. Note that our q-geometric Dirac operators are not exactly parts of spectral triples in the strict Connes sense, although the one on the q-sphere comes close at the algebraic level. The bounded commutator issue was already noted at the end of Sect. 5 for the q-deformed D in (7.15). In [KS12] it is shown for SU q 2 that an operator with similar eigenvalues to the above D has bounded 'commutator' provided the latter is twisted by the left modular automorphism. Investigation of the precise relationship between algebro-geometric triples, such as in [Maj03,BM15], and twisted spectral triples [Con08], should be an interesting topic for further work.
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