PENTAGON EQUATION AND COMPACT QUANTUM SEMIGROUPS M.A.AUKHADIEV AND V.H.TEPOYAN Abstract. The generalization of multiplicative unitary notion 1 from compact quantum groups to compact quantum semigroups 1 0 is considered. We show why the same construction doesn’t work 2 in this case by giving examples of C*-algebraswith non-trivialco- c multiplicationwhichdonotadmitmultiplicativeunitaries. Bythe e use of the pentagon equation we suggest a notion of an operator D which gives comultiplication on any C*-algebra. The multiplica- tive unitary turns out to be its special case. We prove for some 3 1 compactquantumsemigroupsthatthecomultiplicationisgivenby such operator. ] A Q . 1. Introduction h t a The theory of multiplicative unitaries for compact quantum groups m is a well-developed area of the theory of operator algebras. Given the [ importance and the success of that theory, it is natural to attempt 1 to extend it to a more general situation by, for example, developing v a theory of multiplicative unitaries for compact quantum semigroups. 9 0 However, in this work we show that there are significant differences 9 and explain in details why the theory can not be repeated for compact 2 . quantum semigroups. The analogues of the classical results turn out 2 1 to be false. 1 Baaj and Skandalis proved in [4] that there exists a multiplicative 1 unitary u ∈ B(H ⊗H) for every compact quantum group (A,∆) such : v that i X ∆(a) = u(a⊗1)u∗ for any a ∈ A. (1.1) r a ThewaytheyconstructthisoperatorisbasedontheGNS-representation of A corresponding to the Haar functional h of (A,∆), which existence was proved in [3]. Woronowicz proved that a unique faithful Haar functional corresponds to each compact quantum group. In the proof of this theorem the density conditions are used essentially. Hence, this statement may fail to be true for compact quantum semigroups. So, the first difference with the case of compact quantum semigroups is the fact that h may not exist. 1991 Mathematics Subject Classification. Primary 46L05, 46L65; Secondary 16W30. Key words and phrases. Hopf algebra,compact quantum group, compact quan- tum semigroup,multiplicative unitary, pentagon equation, Toeplitz algebra. 1 2 M.A.AUKHADIEVANDV.H.TEPOYAN One can suggest that the theory should be developed for compact quantum semigroups with Haar functional. But the second difference is that even if h exists, it may not be faithful. This is the case for the compact quantum semigroup on the Toeplitz algebra [1] and the algebra of continuous functions on a compact semigroup, which we de- scribe in Section 3. In the first example, despite that h is not faithful, thecorrespondingGNS-representationofthisalgebraisstill isomorphic to itself. But, in Section 3 we prove that there does not exist multi- plicative unitary satisfing (1.1). Nevertheless, there exists a unitary u, which satisfies (1.1), but is not a multiplicative unitary. It follows that multiplicative unitary does not correspond to every compact quantum semigroup. The second example shows us another situation. The Haar func- tional for this compact quantum semigroup exists, and even there is a multiplicative unitary. But, due to the fact that h is not faithful, the corresponding GNS-representation is one-dimensional. This im- plies that the comultiplication defined by the multiplicative unitary does not induce the comultiplication on the algebra as in the classical case. All these reasons bring us to the need for a new notion of operator, which could generalize the notion of a multiplicative unitary. This new operator must (1) not depend on the Haar functional; (2) induce the comultiplication on the algebra; (3) be based on the pentagon equation; (4) give a mutiplicative unitary in the case of compact quantum group; (5) exist for all examples mentioned above. In Section 4 we suggest a notion of an operator which posesses all these properties. Futhermore, we show the way it is defined for the examples from Section 3. We also prove that this operator exists for compact quantum semigroups with some additional properties. The authors are thankful to A. van Daele for discussing some ques- tions of this work with one of the authors. 2. Preliminaries A unital C∗-algebra A with unital *-homomorphism ∆: A → A⊗A iscalledacompact quantum semigroup ([2])if∆satisfiescoassociativity condition: (∆⊗id)∆ = (id⊗∆)∆. ∆ is called a comultiplication. If the linear subspaces {∆(b)(a⊗I); a,b ∈ A}, (2.1) {∆(b)(I ⊗a); a,b ∈ A}, (2.2) PENTAGON EQUATION AND COMPACT QUANTUM SEMIGROUPS 3 are dense in A⊗A, then (A,∆) is called a compact quantum group [3]. A *-homomorphism ǫ: A → C is called a counit if for any a ∈ A (ǫ⊗id)∆(a) = a, (id⊗ǫ)∆(a) = a. State h ∈ A∗ is called a Haar functional in A∗ if the following conditions hold for any ρ ∈ A∗: h∗ρ = ρ∗h = λ h, (2.3) ρ where λ ∈ C depends on ρ. The operation ∗ is a multiplication in A∗ ρ induced by comultiplication ∆ on A: (ρ∗ϕ)(a) = (ρ⊗ϕ)∆(a). Remark 1. h is Haar functional iff the following relations hold for any a ∈ A (h⊗id)∆(a) = h(a)I, (2.4) (id⊗h)∆(a) = h(a)I. (2.5) Further we recall the definition of multiplicative unitary and give a new definition of multiplicative isometry. Definition 1. Let H be a Hilbert space, σ : H ⊗H → H ⊗ H a flip operator: σ(x⊗y) = y ⊗x, where x,y ∈ H. For a ∈ B(H ⊗H) we denote a = a⊗I,a = I ⊗a,a = σ a σ = σ a σ . (2.6) 12 23 13 12 23 12 23 12 23 Operator u ∈ B(H ⊗H) is called multiplicative isometry, if it is isometric and satisfies pentagon equation: u u u = u u . (2.7) 12 13 23 23 12 Ifintheabovedefinitionoperatoruisunitaryitiscalledamultiplica- tive unitary [4]. Recall the construction of multiplicative unitary for a compact quantum group. Consider the GNS-representation (H ,π ) ϕ ϕ of C∗-algebra A, corresponding to a state ϕ. Denote by N a left ideal ϕ {a ∈ A| ϕ(a∗a) = 0}. For any a ∈ A denote by a the corresponding equivalence class in the quotient space A/N . ϕ The main result concerning multiplicative unitary for compact quan- tum groups, proved in [4], is the following. Theorem 1. Let (A,∆) be a compact quantum group with Haar func- tional h. Consider the GNS-representation (H ,π ) corresponding to h h h. Then there exists multiplicative unitary u ∈ B(H ⊗H ) such that h h (π ⊗π )∆(a) = u(π (a)⊗I)u∗. (2.8) h h h For any a,b ∈ A operator u here is defined in the following way u a⊗b = ∆(a)(I ⊗b). (2.9) (cid:0) (cid:1) 4 M.A.AUKHADIEVANDV.H.TEPOYAN 3. Multiplicative unitary and compact quantum semigroups Inthissectionweattempttorepeattheconstructionofmultiplicative unitary for compact quatum semigroups. Firstly we give an analogue of Theorem 1. Theorem 2. Let (A,∆) be a compact quantum semigroup with Haar functional h and (H ,π ) the GNS-representation corresponding to h. h h Then there exists multiplicative isometry u ∈ B(H ⊗H ) such that h h u∗(π ⊗π )(∆(a))u = π (a)⊗I. (3.1) h h h Proof. Take operator u defined by (2.9). Since h is Haar functional, using (2.4) we obtain u a⊗b ,u a⊗b = (h⊗h)((I ⊗b∗)∆(a∗)∆(a)(I ⊗b)) = (cid:10) (cid:0) (cid:1) (cid:0) (cid:1)(cid:11) = h((h⊗id)((I ⊗b∗)∆(a∗a)(I ⊗b))) = h(b∗(h⊗id)∆(a∗a)b) = = h(b∗b)h(a∗a) = a⊗b,a⊗b (cid:10) (cid:11) Consequently, u is isometric. The following calculations show that u is a multiplicative isometry. u u u a⊗b⊗c = u u a⊗u b⊗c = 12 13 23 12 13 (cid:0) (cid:1) (cid:0) (cid:0) (cid:1)(cid:1) = u u a⊗∆(b)(I ⊗c) = u u ((a⊗∆(b))(I ⊗I ⊗c)) = 12 13(cid:16) (cid:17) 12 13 = (∆⊗id)(∆(a))(id⊗∆)(I ⊗b)(I ⊗I ⊗c) = = (id⊗∆)(∆(a)(I ⊗b))(I ⊗I ⊗c) = u u a⊗b⊗c . 23 12 (cid:0) (cid:1) It is easy to check (3.1). (cid:3) Since the density conditions may not hold for an arbitrary compact quantum semigroup, operator u from theorem 2 may not be unitary. Hence, we cannot go further and get the same result as in theorem 1. Next we give examples to describe the situation in details. Example 1. Toeplitz algebra. Consider the Toeplitz algebra T – the minimal C∗-algebra generated byanisometric right-shiftoperatorT andT∗ onaHilbert spaceH with ∞ orthonormal basis {e } . It was shown in[1] that this algebra admits n n=0 a comultiplication ∆ and the corresponding Haar functional h, defined by: ∆(T) = T ⊗T,h(I) = 1, h(T ) = 0, T = TnTm∗ (3.2) n,m n,m for all (m,n) ∈ (Z × Z ). Here we use notation introduced in [1]. + + Obviously, functional h is not faithful. PENTAGON EQUATION AND COMPACT QUANTUM SEMIGROUPS 5 Proposition 1. The GNS-representation of algebra T corresponding to the Haar functional h is the same Toeplitz algebra T , i.e. we have the isomorphism T ≈ π (T) with π (T) being right-shift operator on h h H . h Proof. OnecaneasilyseethatN isalinearspacegeneratedby{T } . h n,m m6=0 Here H is a Hilbert space with basis {e }∞ , where e = Tk = k k=0 k (cid:2) (cid:3) Tk + N . Then the corresponding GNS-representation π acts in the h h following way: π (Tn)e = π (Tn) Tk = Tn+k = e . h k h n+k (cid:2) (cid:3) (cid:2) (cid:3) Thus, the algebra π (T) is a C∗-algebra generated by right-shift iso- h metric π (T). h (cid:3) Theorem 3. There doesn’texistanymultiplicativeunitary ufor(T ,∆) satisfying (2.8). But there exists a unitary operator u satisfying (2.8) which doesn’t satisfy pentagon equation. Proof. By virtue of proposition 1 we may identify π (T ) with T and h omit notation π . Assume that u ∈ B(H ⊗ H) is a unitary operator h satisfying (2.8) for all a ∈ A. Particularly, we have T⊗T = u(T⊗1)u∗. Hence, such operatoris equivalent toonedefined by following relations: u(e ⊗e ) = e ⊗e , k = 0,1,2,... , 0 2k 0 k u(e ⊗e ) = e ⊗e , k = 1,2,... . 0 2k−1 k 0 It is sufficient to show that pentagon equation does not hold for this operator. To this end take vector e ⊗ e ⊗ e and calculate first the 0 1 0 left-hand side of pentagon equation. u u u (e ⊗e ⊗e ) = u u (e ⊗u(e ⊗e )) = u u (e ⊗u(T⊗1)(e ⊗e )) = 12 13 23 0 1 0 12 13 0 1 0 12 13 0 0 0 = u u (e ⊗(T⊗T)u(e ⊗e )) = u u (e ⊗e ⊗e ) = u (e ⊗e ⊗e ) = 12 13 0 0 0 12 13 0 1 1 12 1 1 0 = (u(T ⊗1)(e ⊗e )⊗e ) = ((T ⊗T)u(e ⊗e )⊗e ) = e ⊗e ⊗e . 0 1 0 0 1 0 2 1 0 And the right-hand side of pentagon equation on the same vector. u u (e ⊗e ⊗e ) = u (e ⊗e ⊗e ) = e ⊗e ⊗e . 23 12 0 1 0 23 1 0 0 1 0 0 Consequently, this operator u is not a multiplicative unitary. (cid:3) Thus, there arecompact quantum semigroups without multiplicative unitary. Next example describes another kind of situation. Example 2. Compact semigroup algebra 6 M.A.AUKHADIEVANDV.H.TEPOYAN Let S be a compact semigroup with zero element. Consider the algebra of continious functions on S, C(S). Define the natural comul- tiplication on C(S). (∆(f))(x,y) = f (xy). (3.3) One can easily verify that (C(S),∆) is a compact quantum semi- group. The Haar functional on (C(S),∆) is the next one: h(f) = f (0) Obviously, h is not faithful. Since h is a pure state, the corresponding GNS-representation of C(S) is one-dimensional. Therefore, there is no interest in considering multiplicative unitary in this example. In the next section we give the definition of operator based on pen- tagon equation. This operator generalizes the notion of multiplicative unitary and is sufficient for these two examples. 4. New operator ThenotionofmultiplicativeunitaryisbasedontheGNS-representation corresponding to Haar functional h. Since the representation π is h faithful for a compact quantum group (A,∆), the comultiplication ∆ ′ induces comultiplication ∆ on π (A), defined by the following com- h mutative diagram: ∆ A −−−→ A⊗A πh πh⊗πh   y ′ y ∆ π (A) −−−→ π (A)⊗π (A) h h h In the case of compact quantum group we identify A with π (A). h The multiplicative unitary from theorem 1 then satisfies the next con- dition ∆′ (a) = u∗(a⊗1)u (4.1) for any a ∈ A. The right-hand side is the unitary operator W defined as multipli- cation by u∗ from the left, and u from the right, calculated on a⊗1, the element of π (A)⊗π (A), which we identify with A. This shows h h that W : A⊗A → A⊗A is a linear unitary operator. The pentagon equation for operator u is encoded in the similar equation for operator W. We can rewrite (4.1) in the following way: ′ ∆ (a) = W(a⊗1). We have already shown that there may not exist (see ex. 1) a multi- plicativeunitary, oritmaygivenointerestsincetheGNS-representation may not be faithful (see ex. 2). Neverthless, the operator W described above may still exist and may give the comultiplication, without being PENTAGON EQUATION AND COMPACT QUANTUM SEMIGROUPS 7 unitar. This leads us to an idea of an operator W : A⊗A → A⊗A which satisfies the pentagon equation on algebra. Let L(A) be the algebra of all linear continuous operators on C∗- algebra A. Operator Σ ∈ L(A⊗A) defined as follows Σ(a⊗b) = b ⊗ a, for a,b ∈ A, is called a flip. Suppose that V ∈ L(A⊗A) and a ∈ A⊗A, then we denote V = V ⊗id, V = id⊗V, V = Σ V Σ = Σ V Σ 12 23 13 12 23 12 23 12 23 a = a⊗1, a = 1⊗a, a = (Σ⊗id)(a ) = Σ a 12 23 13 23 12 23 Definition 2. Let A be a C∗-algebra. We say that the linear opera- tor W : A ⊗ A → A ⊗ A satisfies pentagon equation if the following condition holds W W W = W W . (4.2) 12 13 23 23 12 Given a unital C∗-algebra A we may define two trivial comultiplica- tions on it: ∆L(a) = a⊗1 ∆R(a) = 1⊗a Then A,∆L and A,∆R are compact quantum semigroups. (cid:0) (cid:1) (cid:0) (cid:1) Theorem 4. Let A be a unital C∗-algebra and W ∈ L(A⊗A) be a unital C∗-homomorphism, which satisfies pentagon equation. Then op- erators ∆(a) = W∆(a) and ∆(a) = W∗∆R(a) define comultiplications on A and (A,∆), (A,∆) arebcompact quantum semigroups. b Proof. It is enough to check the coassociativity of maps ∆,∆. Take a ∈ A⊗A. Then we have b (∆⊗id)a = W∆L ⊗id a = (cid:0) (cid:1) = (W ⊗id) ∆L ⊗id a = (W ⊗id)a = W (a ) 13 12 13 (cid:0) (cid:1) Using (4.2) and the above relation we get (∆⊗id)∆(a) = W (∆(a)) = W (W (a⊗1)) = 12 13 12 13 W W (a⊗1) = 12 13 13 = W W (a⊗1⊗1) = W W (id⊗W)(a⊗1⊗1) = 12 13 12 13 = W W W (a⊗1⊗1) = W W (a⊗1⊗1) = 12 13 23 23 12 W (W (a⊗1)) = 23 12 W (∆(a)) = (id⊗∆)∆(a). 23 12 (cid:3) By the similar calculations we obtain the same for ∆. b Theorem 5. For any compact quantum semigroup (A,∆) with counit ǫ, there exist C∗-homomorphisms WL,WR ∈ L(A⊗A) satisfying the pentagon equation, such that ∆ = WL∆L = WR∆R. 8 M.A.AUKHADIEVANDV.H.TEPOYAN Remark 2. Under conditions of Theorem 5 WL,WR are projections and satisfy the following conditions WLWR = WR,WRWL = WL. ′ ′ Moreover, there exist C∗-homomorphisms WL ,WR ∈ L(A⊗A), ′ ′ WL = (id⊗ǫ(·)I),WR = (ǫ(·)I ⊗id) which are also projections such that ′ ′ WL ∆ = ∆L,WR ∆ = ∆R ′ ′ WLWL = WL,WRWR = WR. Consider two operators V ∈ L(A⊗A), W ∈ L(B⊗B) satisfying pentagon equation. Define linear operator V ⊠W = (id⊗Σ⊗id)(V ⊗W)(id⊗Σ⊗id). Clearly, V ⊠W satisfies pentagon equation on A⊗B ⊗A⊗B. Theorem 6. Let (A,∆ ), (B,∆ ) be compact quantum semigroups A B and W ∈ L(A⊗A),W ∈ L(B⊗B) be operators satisfying penta- A B gon equation, such that ∆ = W ∆ , ∆ = W ∆ . A A 1 B B 1 Then (A⊗B,∆) is a compact quantum semigroup with ∆ given by: ∆ = (W ⊠W )∆ A B 1 References [1] M.A.Aukhadiev, S.A.Grigoryan, E.V.Lipacheva: A Compact Quantum Semi- group Generated by an Isometry, Russian Mathematics (Iz. VUZ), Vol.55, No.10, pp. 78–81,2011. [2] A. Maes, A. Van Daele.: Notes on Compact Quantum Groups, Nieuw Arch. Wisk. 4 16, no. 1-2, 73–112 (1998) [3] S.L. Woronowicz.: Compact quantum groups, Sym´etries quantiques (Les Houches, 1995), 845–884,North-Holland, Amsterdam, 1998. [4] Baaj S., Skandalis G.:Unitaires multiplicatifs et dualit´e pour les produits crois´es de C*-alg´ebres, Ann. Scient. Ec. Norm. Sup., 4e s´erie, 26, 425 488 (1993). (M.A.Aukhadiev,V.H.Tepoyan)KazanStatePowerEngineeringUniver- sity, Krasnoselskaya str., 51, 420066, Kazan, Russia E-mail address, M.A.Aukhadiev: [email protected] E-mail address, V.H.Tepoyan: [email protected]