Rapid decay and polynomial growth for bicrossed products

We study the rapid decay property and polynomial growth for duals of bicrossed products coming from a matched pair of a discrete group and a compact group


Introduction
In the breakthrough paper paper [Ha78], Haagerup showed that the norm of the reduced C*-algebra C * r (F N ) of the free group on N -generators F N , can be controlled by the Sobolev l 2 -norms associated to the word length function on F N .This is a striking phenomenon which actually occurs in many more cases.Jolissaint recognized this phenomenon, called Rapid Decay (or property (RD)), and studied it in a systematic way in [Jo90].Property (RD) has now many applications.Let us mention the remarkable one concerning Ktheory.Property (RD) allowed Jolissaint [Jo89] to show that the K-theory and C * r (Γ) equals the K-theory of subalgebras of rapidly decreasing functions on Γ (Jolissaint did attribute this result to Connes).This result was then used by V. Lafforgue in his approach to the Baum-Connes conjecture via Banach KK-theory [La00,La02].
In this paper, we view discrete quantum groups as duals of compact quantum groups.The theory of compact quantum groups has been developed by Woronowicz [Wo87,Wo88,Wo98].Property (RD) for discrete quantum groups has been introduced and studied by Vergnioux [Ve07].Property (RD) has been refined later [BVZ14] in order to fit in the context of non-unimodular discrete quantum groups.
In this paper, we study the permanence of property (RD) under the bicrossed product construction.This construction was initiated by Kac [Ka68] in the context of finite quantum groups and was extensively studied later by many authors in different settings.The general construction, for locally compact quantum groups, was developed by Vaes-Vainerman [VV03].In the context of compact quantum groups given by matched pairs of classical groups, an easier approach, that we will follow, was given by Fima-Mukherjee-Patri [FMP17].
Following [FMP17], the bicrossed product construction associates to a matched pair (Γ, G) of a discrete group Γ and a compact group G (see Section 2.2) a compact quantum group G, called the bicrossed product.Given a length function l on the set of equivalence classes Irr(G) of irreducible unitary representations of G one can associate in a canonical way, as explained in Proposition 4.2, a pair of length functions (l Γ , l G ) on Γ and Irr(G) respectively.Such a pair satisfies some compatibility relations and every pair of length functions (l Γ , l G ) on (Γ, Irr(G)) satisfying those compatibility relations will be called matched (see Definition 4.1).Any matched pair (l Γ , l G ) on (Γ, Irr(G)) allows one to reconstruct a canonical length function on Irr(G).The main result of the present paper is the following.
Theorem A. Let (Γ, G) be a matched pair of a discrete group Γ and a compact group G. Denote by G the bicrossed product.The following are equivalent.
For amenable discrete groups, property (RD) is equivalent to polynomial growth [Jo90] and the same occurs for discrete quantum groups [Ve07].Hence, for the compact classical group G one has that ( G, l G ) has (RD) if and only if it has polynomial growth.Note that a bicrossed product of a matched pair (Γ, G) is co-amenable if and only if Γ is amenable [FMP17].The following theorem shows the permanence of polynomial growth under the bicrossed product construction.
Theorem B. Let (Γ, G) be a matched pair of a discrete group Γ and a compact group G. Denote by G the bicrossed product.The following are equivalent.
2. There exists a matched pair of length function (l Γ , l G ) on (Γ, Irr(G)) such that both (Γ, l Γ ) and ( G, l G ) have polynomial growth.
The main ingredient to prove Theorem A and B is the classification of the irreducible unitary representation of a bicrossed product and the fusion rules.
The paper is organized as follows.Section 2 is a preliminary section in which we introduce our notations.In section 3 we classify the irreducible unitary representation of a bicrossed product and describe their fusion rules.Finally, in section 4, we prove Theorem A and Theorem B.

Compact bicrossed products
In this section, we follow the approach and the notations of [FMP17].
Let (Γ, G) be a pair of a countable discrete group Γ and a second countable compact group G with a left action α : Γ → Homeo(G) of Γ on the compact space G by homeomorphisms and a right action β : G → S(Γ) of G on the discrete space Γ, where S(Γ) is the Polish group of bijections of Γ, the topology being the one of pointwise convergence i.e., the smallest one for which the evaluation maps S(Γ) → Γ, σ → σ(γ) are continuous, for all γ ∈ Γ, where Γ has the discrete topology.Here, α is a group homomorphism and β is an antihomomorphism.The pair (Γ, G) is called a matched pair if Γ ∩ G = {e} with e being the common unit for both G and Γ, and if the actions α and β satisfy the following matched pair relations: 3 Representation theory of bicrossed products

Classification of irreducible representations
In this section we classify the irreducible representations of a bicrossed product.Let (Γ, G) be a matched pair of a discrete countable group Γ and a second countable compact group G with actions α, β.
For γ ∈ Γ we denote by G γ := G γ,γ the stabilizer of γ for the action β : Γ G.Note that G γ is an open (hence closed) subgroup of G, hence of finite index: its index is |γ •G|.We view C(G γ ) = v γγ C(G) ⊂ C(G) as a non-unital C*-subalgebra.Let us denote by ν the Haar probability measure on G and note that ν(G γ ) = 1 |γ•G| so that the Haar probability measure ν γ on G γ is given by ν γ (A) = |γ • G| ν(A) for all Borel subset A of G γ .
For γ ∈ Γ we fix a section, still denoted γ, γ : This means that γ : γ •G → G is an injective map such that γ •γ(r) = r for all r ∈ γ •G.We choose the section γ such that γ(γ) = 1, for all γ ∈ Γ.For r, s ∈ γ • G, we denote by ψ γ r,s the ν-preserving homeomorphism of G defined by ψ γ r,s (g) = γ(r)gγ(s) −1 .It follows from our choices that ψ γ γ,γ = id for all γ ∈ Γ.Moreover, for all g ∈ G, one has ψ γ r,s (g) ∈ G γ if and only if g ∈ G r,s .It follows that ψ γ r,r is an isomorphism and an homeomorphism from G r to G γ intertwining the Haar probability measures.
Let u : G γ → U(H) be a unitary representation of G γ and view u as a continuous function G → B(H) which is zero outside G γ i.e. a partial isometry in B(H) ⊗ C(G) such that uu * = u * u = id H ⊗ v γγ .Define, for r, s ∈ γ • G, the partial isometry u r,s := u • ψ γ r,s := (g → u(ψ γ r,s (g))) ∈ B(H) ⊗ C(G) and note that u * r,s u r,s = u r,s u * r,s = id H ⊗ 1 Gr,s .In the sequel we view u r,s ∈ B(H) ⊗ C(G) ⊂ B(H) ⊗ C(G) and we define: where we recall that e rs , for r, s ∈ γ • G, are the matrix units associated to the canonical orthonormal basis of l 2 (γ • G).
The irreducible unitary representations of G are described as follows.

The character of
G γ is a group isomorphism and an homeomorphism).
5. γ(u) is irreducible if and only if u is irreducible.Moreover, for any irreducible unitary representation u of G there exists γ ∈ Γ and v an irreducible representation of G γ such that u ≃ γ(v).
Proof.(1).Writing γ(u) = r,s e r,s ⊗ V r,s , where V r,s := (1 ⊗ u r v rs )u r,s ∈ B(H) ⊗ C(G), it suffices to check that, for all r, s ∈ γ • G one has (id ⊗ ∆)(V r,s ) = t∈γ•G (V r,t ) 12 (V t,s ) 13 .We first claim that, for all r, s ∈ γ • G, (id ⊗ ∆)(u r,s ) = t∈γ•G (u r,t ) 12 (u t,s ) 13 .To check our claim, first recall that, for all r, s ∈ γ • G one has Since we also have u t,s (h) = 0 whenever r • gh = s we find, in both cases, that u r,s (gh) = u r,t (g)u t,s (h).Now, for t = r • g we have u r,t (g) = 0 so the following formulae holds for any r, s ∈ γ • G and any g, h ∈ G: Hence, for all r, s, t ) 12 (u t,s ) 13 .Using this we find: Since v γ is a unitary representation of G and a magic unitary we also have: This shows that γ(u) is a representation of G.We now check that γ(u) is unitary.As before, since for all (3).Let γ, µ ∈ Γ and u, w be representations of G γ and G µ respectively.Since the Haar measure on G is invariant under the action α and the homeomorphisms ψ γ r,r and ψ µ r,r , we find, by the character formulae in 2 and the crossed-product relations, Hence, h ∈ G µ .Since the characters of finite dimensional unitary representation of a group Λ are central functions i.e. invariant under Ad(λ) for all λ ∈ Λ, we have χ(w) (4).Note that, by the bicrossed product relations, we have, for all γ ∈ Γ and g ∈ G, (γ is an homeomorphism and, by the bicrossed product relations, one has, for all g ∈ G γ and h ∈ G, α γ (gh ).This implies that, for all γ ∈ Γ, there exists a map Let now r ∈ γ • G and g ∈ G r .One has, using the bicrossed product relations, that e = α r (γ(r)γ(r Since, as we seen above, It then follows from the crossedproduct relations and the discussion above : (5).The statement on irreducibility following from 3, it suffices, by the general theory, to show that the linear span X of coefficients of representations of the form γ(u), for γ ∈ Γ and u an irreducible unitary representation of G γ , is a dense subset of C(G).Note that, for all γ ∈ Γ, the relation 1 , where F ∈ C(G γ ).Since the linear span of coefficients of irreducible unitary representation of G γ is dense in C(G γ ), it suffices to show that, for any γ ∈ Γ, for any irreducible unitary representation of Finally, the fusion rules are described as follows.
3. For all γ 1 , γ 2 , γ 3 ∈ Γ and all u, v, w unitary representations of G γ1 , G γ2 and G γ3 respectively, the number ) is equal to: Let us observe that, by the bicrossed product relations, we have, for all γ 1 , γ 2 , γ 3 ∈ Γ, Proof.(1).Put w = u ⊗ r v and let g, h ∈ G r .Then, w(gh) is equal to: Since v ty (g) = 0 precisely when t • g = y, the factor Moreover, since for all g ∈ G r and all s, t such that st = r, one has, whenever t • g = y and s it follows that the only non-zero terms in the last sum are for x ∈ γ • G and y ∈ µ • G such that xy = r.By the properties of the matrix units we see immediately that w(gh) = w(g)w(h).To end the proof of (1), it suffices to check that w(1) = 1, which is clear, and that w(g) * = w(g −1 ) for all g ∈ G r .So let g ∈ G r .One has: Note that for all t, t ′ ∈ Γ and all g ∈ G, one has v s ′ s (g) = v ss ′ (g −1 ).Also, using the bicrossed product relations one finds that α r (g) −1 = α r•g (g −1 ) for all r ∈ Γ and g ∈ G.In particular, ).It follows immediately that w(g) * = w(g −1 ).
(3).One has dim(Mor G (γ Note that, whenever there is no non-zero terms in the sum above.

The induced representation
In this section, we explain how the induced representation maybe viewed as a particular twisted tensor product.
For γ ∈ Γ and u : G γ → U(H) is a unitary representation of G γ we define the induced representation: e rs ⊗ v rs (g)u(ψ γ rs (g)).
It follows from Theorem 3.2 that Ind G γ (u) is indeed a unitary representation of G.We collect some elementary and well known facts about this representation in the following Proposition.Note that, in property 3, we use the symbol Res G Gγ (u) for u ∈ Rep(G) to denote the restriction of u to a representation of G γ .Hence, property 3 motivates the name induced representation for the representation Ind G γ (u).
1.For all γ ∈ Γ and all 2. For all γ ∈ Γ and all

Proof. (1). It is obvious, by definition of Ind
Since ψ γ rr : G r → G γ is a Haar probability preserving homeomorphism we obtain Finally, since, for all g ∈ G, χ(u)

Length functions
Recall that given a compact quantum group H, a function l : and that l(x) ≤ l(y) + l(z) whenever x ⊂ y ⊗ z.A length function on a discrete group Λ is a function l : Λ → [0, ∞) such that l(1) = 0, l(r) = l(r −1 ) and l(rs) ≤ l(r) + l(s) for all r, s ∈ Λ.
Let (Γ, G) be a matched pair with bicrossed product G.In view of the description of the irreducible representations of G, the fusion rules and the contragredient representation, it is clear that to get a length function on Irr(G), we need a family of maps l γ : Irr(G γ ) → [0, +∞[, for γ ∈ Γ, satisfying the hypothesis of the following definition.(iii) For any γ ∈ Γ, x ∈ Irr(G γ ), we have for some r ∈ γ 3 • G, then l γ3 (z) ≤ l γ1 (x) + l γ2 (y).(4.2) The next Proposition shows that our notion of matched pair for length functions is the good one, as expected.

If l is a length function on Irr(G) then the maps l
Finally, note that, by point 4 of Theorem 3.1, for all γ ∈ Γ, one has So l Γ is a length function on Γ.It is obvious that l G is a length function on Irr(G).Let us prove that the pair (l Γ , l G ) is matched.Indeed, defining l γ : Next, by point 4 of Theorem 3.1, we have [γ(u which proves point (ii) of Definition 4.1.Finally, for point (iv), the fusion rules in Theorem 3.2 imply dim Mor(γ 3 (u z ), γ 1 (u If dim Mor Gr (u z • ψ γ3 r,r , u x ⊗ r u y ) = 0 for some r ∈ γ 3 • G, then (4.3) is also nonzero, which means, by irreducibility of Hence, since l is a length function on Irr(G), (2).Since l Γ is β-invariant, the map l ′ is well defined by Theorem 3.1.It is clear that l ′ (ε G ) = 0 and, by point 4 (and 5) of Theorem 3.1 and since l ′ is a length function we also have that l ′ (z) = l ′ (z) for all z ∈ Irr(G).Let now γ 1 , γ 2 , γ 3 ∈ Γ, x ∈ Irr(G γ1 ), y ∈ Irr(G γ2 ) and z ∈ Irr(G γ3 ) be such that γ 1 (u x ) ⊂ γ 2 (u y ) ⊗ γ 3 (u z ) then, by point 3 in Theorem 3.2, there exists (3).Let (l Γ , l G ) be a matched pair of length functions.By points 1 and 2 of Definition 4.1 we have, for all γ ∈ Γ and all Hence, l Γ is β-invariant.By assertion (2) we just proved above, we get a length function l ′ on Irr(G).Now, it is clear from Definition 4.1, the fusion rules and the adjoint representation of a bicrossed product (point 3 of Theorem 3.2 and point 4 of Theorem 3.1) that l : [γ(u x )] → l γ (x) is a length function on Irr(G).Since l = l + l ′ , l is also a length function on Irr(G).

Rapid decay and polynomial growth
In this section we study property (RD) and polynomial growth for bicrossed-products.

Generalities
We use the notion of property (RD) developed by Vergnioux in [Ve07] (see also [BVZ14]) and recall the definition below.Since we are only dealing with Kac algebras, we recall the definition of the Fourier transform and rapid decay only for Kac algebras.
Let H be a compact quantum group.We use the notation l ∞ ( H) := x∈Irr(H) B(H x ) to denote the l ∞ direct sum.The c 0 direct sum is denoted by c 0 ( H) ⊂ l ∞ ( H) and the algebraic direct sum is denoted by c c ( H) ⊂ c 0 ( H).An element a ∈ c c ( H) is said to have finite support and its finite support is denoted by Supp(a) := {x ∈ Irr(H) : ap x = 0}, where p x , for x ∈ Irr(H) denotes the central minimal projection of l ∞ ( H) corresponding to the block B(H x ).
For a compact quantum group H which is always supposed to be of Kac type, and a ∈ C c ( H) we define its Fourier transform as: and its "Sobolev 0-norm" by a 2 H,0 = x∈Irr(H) dim(x)Tr x ((a * a)p x ).Given a length function l : Irr(H) → [0, ∞), consider the element L = x∈Irr(H) l(x)p x which is affilated to c 0 ( H).Let q n denote the spectral projections of L associated to the interval [n, n + 1).
The pair ( H, l) is said to have: • Property (RD) if there exists a polynomial P ∈ R[X] such that for every k ∈ N and a ∈ q k c c ( H), we have F (a) C(H) ≤ P (k) a H,0 .
Finally, H is said to have polynomial growth (resp.property (RD) if there exists a length function l on Irr(H) such that ( H, l) has polynomial growth (resp.property (RD)).
It is known from [Ve07] that if ( H, l) has polynomial growth then ( H, l) has rapid decay and the converse also holds when we assume H to be co-amenable.Moreover, it is shown also shown in [Ve07] that duals of compact connected real Lie groups have polynomial growth.The fact that polynomial growth implies (RD) can easily be deduced from the following lemma.
Lemma 5.1.Let H be a CQG, F ⊂ Irr(H) a finite subset and a ∈ l ∞ ( H) with ap x = 0 for all x / ∈ F .Then, Proof.One can copy the proof of Proposition 4.2, assertion (a), in [BVZ14] or the proof of Proposition 4.4, assertion (ii), in [Ve07].

Technicalities
Let (Γ, G) be a matched pair with actions (α, β) and denote by G the bicrossed product.
Recall that Irr(G) = ⊔ γ∈I Irr(G γ ), where I ⊂ Γ is a complete set of representatives for Γ/G.For γ ∈ I and x ∈ Irr(G γ ), we denote by γ(x) the corresponding element in Irr(G).If a complete set of representatives of Irr(G γ ), x ∈ Irr(G γ ) is given by u x ∈ B(H x ) ⊗ C(G γ ) then a representative for γ(x) is given by The lemma below gives a way of obtaining an element a ∈ c c ( G) from an a ∈ c c ( G γ ) in a suitable way so that they are compatible with the Fourier transforms.
Lemma 5.2.Let γ ∈ Γ and a ∈ c c ( G γ ).Define a ∈ c c ( G) by: where S y i ∈ Mor(y, Ind G γ (x)) is a basis of isometries with pairwise orthogonal images.The following holds.
1.If (l Γ , l) is a matched pair of length functions on (Γ, Irr(G)) then, for all y ∈ supp( a) one has where (l γ ) γ∈Γ is any family of maps realizing the compatibility relations of Definition 4.1.
(2).One has: where, in the 3rd equation we use the fact that (S y i ) * ∈ Mor(Ind G γ (x), y) and, in the last equation we use the definition of the representation Ind G γ (u x ).(3).One has: dim(x)Tr x (a * ap x ) = a 2 Gγ,0 .

Polynomial growth for bicrossed product
We start with the following result.
To complete the proof of Theorem B, we need the following Proposition.
Proposition 5.5.Assume that there exists a length function l on Irr(G) such that ( G, l) has polynomial growth and consider the matched pair of length functions (l Γ , l G ) associated to l given in Proposition 4.2.Then (Γ, l Γ ) and ( G, l G ) both have polynomial growth.
To complete the proof of Theorem A, we need the following Proposition.This concludes the proof.

For a
Hilbert space H, we denote by U(H) its unitary group and by B(H) the C*-algebra of bounded linear operators on H.When H is finite dimensional, we denote by Tr the unique trace on B(H) such that Tr(1) = dim(H).We use the same symbol ⊗ for the tensor product of Hilbert spaces, unitary representations of compact quantum groups, minimal tensor product of C*-algebras.For a compact quantum group G, we denote by Irr(G) the set of equivalence classes of irreducible unitary representations and Rep(G) the collection of finite dimensional unitary representations.We will often denote by[u]  the equivalence class of an irreducible unitary representation u.For u ∈ Rep(G), we denote by χ(u) its character, i.e., viewing u ∈ B(H) ⊗ C(G) for some finite dimensional Hilbert space H, one has χ(u) := (Tr⊗id)(u) ∈ C(G).We denote by Pol(G) the unital C*-algebra obtained by taking the Span of the coefficients of irreducible unitary representation, by C m (G) the enveloping C*-algebra of Pol(G) and by C(G) the C*-algebra generated by the GNS construction of the Haar state on C m (G).We also denote by ε : C m (G) → C the counit and we use the same symbol ε ∈ Irr(G) to denote the trivial representation and its class in Irr(G).In the entire paper, the word representation means a unitary and finite dimensional representation.
and α γ (e) = β g (e) = e.(2.1)We also write γ • g := β g (γ).From now on, we assume (Γ, G) is matched.It is shown in [FMP17, Proposition 3.2] that β is automatically continuous.By continuity of β and compactness of G, every β orbit is finite.Moreover, the sets G r,s := {g ∈ G : r• g = s} are clopen (see [FMP17, Section 2.1]).Let v rs = 1 Gr,s ∈ C(G) be the characteristic function of G r,s .It is shown in [FMP17, Section 2.1] that, for all β-orbits γ • G ∈ Γ/G, the unitary v γ•G := r,s∈γ•G e rs ⊗ v rs ∈ B(l 2 (γ • G)) ⊗ C(G)is a unitary representation of G as well as a magic unitary, where e rs ∈ B(l 2 (γ • G)) are the canonical matrix units and the Haar probability measure ν on G is α-invariant.It is shown in [FMP17, Theorem 3.4] that there exists a unique compact quantum group G, called the bicrossed product of the matched pair (Γ, G), such that C(G) = Γ α ⋉ C(G) is the reduced C*-algebraic crossed product, generated by a copy of C(G) and the unitaries u γ , γ ∈ Γ and ∆ : C(G) → C(G) ⊗ C(G) is the unique unital * -homomorphism satisfying ∆| C(G) = ∆ G (the comultiplication on C(G)) and ∆(u γ ) = r∈γ•G u γ v γr ⊗ u r for all γ ∈ Γ.It is also shown that the Haar state on G is a trace and is given by the formula h(u γ F ) = δ γ,1 G F dν for all γ ∈ Γ and F ∈ C(G).
by unitary representations of G γ and G µ respectively.For any r ∈ (γ • G)(µ • G), we define the r-twisted tensor product of u and v, denoted u ⊗ r v as a unitary representation of G r on K r ⊗ H u ⊗ H v , where
is a well defined length function on Irr(G).