HERZ-SCHUR MULTIPLIERS

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1 HERZ-SCHUR MULTIPLIERS IVAN. TODOROV Contents 1. Introduction 1 2. Preliminaries Operator spaces Harmonic analysis 4 3. The spaces MA() and M cb A() The case of commutative groups Schur multipliers ω-topology The predual of B(H 1, H 2 ) The space T () The characterisation theorem Discrete and continuous Schur multipliers Further properties of M cb A() Embedding into the Schur multipliers The case of compact groups Coefficients of representations The canonical predual of M cb A() Classes of multipliers Positive multipliers Idempotent multipliers Radial multipliers Approximation properties for groups 50 References Introduction The purpose of these notes is to develop the basics of the theory of Herz- Schur multipliers. This notion was formally introduced in 1985 in [5] and developed by U. Haagerup and his collaborators, as well as by a number of other researchers, in the following decades. The literature on the subject is vast, and its applications far reaching. A major driving force behind these developments were the connections with approximation properties of Date: 27 April

2 2 IVAN. TODOROV operator algebras. In these notes, we will not discuss this side of the subject, and will only briefly mention how Herz-Schur multipliers are used to define approximation properties of the Fourier algebra in Section 7. Instead, we focus on the development of the core material on multipliers on locally compact groups and various specific classes of interest. These notes formed the basis of a series of lectures at the programme Harmonic analysis, Banach and operator algebras at the Fields Institute in March-April 2014; due to time limitations, some aspects of the subject, such as that of Littlewood multipliers, are not included here. 2. Preliminaries 2.1. Operator spaces. We refer the reader to the monographs [8], [41], [42] for background in Operator Space Theory. In this section, we fix notation and include some results that will be used in the sequel. If X is a vector space, we denote as customary by M n (X ) the vector space of all n by n matrices with entries in X. If Y is another vector space and ϕ : X Y is a linear map, we let ϕ (n) : M n (X ) M n (Y) be the map given by ϕ (n) ((x i,j )) = (ϕ(x i,j )); thus, if we identify M n (X ) and M n (Y) with X M n and Y M n, respectively, then ϕ (n) = ϕ id. If H is a Hilbert space and (e i ) i I is a fixed basis, we associate to every element x B(H) its matrix (x i,j ) i,j I. Here, x i,j = (xe j, e i ),i, j I. More generally, if X is an operator space then every element of the spacial norm closed tensor product X min K(H) can be identified with a matrix (x i,j ) i,j I, but this time with x i,j being elements of X. If ϕ : X Y is a completely bounded linear map then there exists a (unique) bounded map ϕ id : X min K Y min K such that ϕ id((x i,j )) = (ϕ(x i,j )). If, moreover, X and Y are dual operator spaces and ϕ is weak*-continuous then there exists a (unique) weak* continuous bounded map ϕ : X B(H) Y B(H) such that ϕ((x i,j )) = (ϕ(x i,j )) for every (x i,j ) i,j I X B(H). Here, denotes the weak* spacial tensor product. The map ϕ will still be denoted by ϕ id. We next include the statement of two fundamental theorems in Operator Space Theory. The first one is Stinespring s Dilation Theorem: Theorem 2.1. Let A be a C*-algebra and Φ : A B(H) be a completely positive map. There exist a Hilbert space K, a non-degenerate *- representation π : A B(K) and a bounded operator V : H K such that Φ(a) = V π(a)v, a A. The second is the Haagerup-Paulsen-Wittstock Factorisation Theorem. Theorem 2.2. Let A be a C*-algebra and Φ : A B(H) be a completely bounded map. There exist a Hilbert space K, a non-degenerate *- representation π : A B(K) and bounded operators V, W : H K such

3 that HERZ-SCHUR MULTIPLIERS 3 Φ(a) = W π(a)v, a A. Moreover, V and W can be chosen so that Φ cb = V W. We next include some results of R. R. Smith [47] and F. Pop, A. Sinclair and R. R. Smith [43] that will be useful in the sequel. Let H be a Hilbert space, A B(H) be a C*-algebra, and X B(H) be an operator space such that AX A X ; such an X is called an A-bimodule. Since the C*-algebra K(H) of all compact operators on H is an ideal in B(H), it is an A-bimodule for every C*-algebra A B(H). Let A be a unital C*-algebra and X be an operator space that is an A- bimodule. We call A matricially norming for X [43] if, for every n N and every X M n (X ), we have that X = sup{ CXD : C = (c 1,..., c n ), D = (d 1,..., d n ) t, C, D 1}. Theorem 2.3. Let A be a unital C*-algebra, X be an A-bimodule and Φ : X X be an A-bimodular map. If Φ is bounded and A is matricially norming for X then Φ is completely bounded with Φ cb = Φ. Proof. We have that Φ (n) = sup{ Φ (n) (X) : X M n (X ) a contraction} = sup{ C Φ (n) (X)D : X M n (X ), C, D M n,1 (A) contractions} = sup{ Φ(C XD) : X M n (X ), C, D M n,1 (A) contractions} Φ. Theorem 2.4. Let H be a Hilbert space, A B(H) be a C*-subalgebra with a cyclic vector. If X B(H) is an A-bimodule then A is matricially norming for X. Proof. Let ξ H be a vector with Aξ = H and let X = (x i,j ) M n (X ) be an operator matrix with X > 1. Then there exist vectors ξ = (ξ 1,..., ξ n ) and η = (η 1,..., η n ) of norm strictly less than 1 such that (Xξ, η ) > 1. Since Aξ = H, there exist elements a i, b i A such that a i ξ (resp. b i ξ) is as close to η i (resp. ξ i ) so that the vectors ξ = (a 1 ξ,..., a n ξ) and η = (b 1 ξ,..., b n ξ) still have norm strictly less than 1 and the inequality (1) (Xξ, η ) > 1 still holds. Let a = n i=1 a i a i and b = n i=1 b i b i. We assume first that a and b are invertible. Let ξ = b 1/2 ξ, η = a 1/2 η, c i = a i a 1/2 and d i = b i b 1/2. Then we have that c i η = a i η and d i ξ = bi ξ, i = 1,..., n, and, by (1), that (2) c i x i,j d j ξ, η i,j > 1.

4 4 IVAN. TODOROV Moreover, ξ 2 = (b 1/2 ξ, b 1/2 ξ) = (bξ, ξ) = n b i ξ 2 < 1, and, similarly, η 2 < 1. It follows that n i,j=1 c i x i,jd j > 1. On the other hand, the operator n i,j=1 c i x i,jd j is equal to the product C(x i,j )D, where C = (c 1,..., c n) and D = (d 1,..., d n ) t. We have that C 2 = n c i c i = i=1 i=1 n a 1/2 a i a i a 1/2 = I = 1 i=1 and, similarly, D = 1. Thus, we have that C(x i,j )D > 1. This completes the proof in the case both a and b are invertible. In case a or b is not invertible, we consider, instead of the vectors ξ and η, the vectors (a 1 ξ,..., a n ξ, ɛξ) H n+1 and (b 1 ξ,..., b n ξ, ɛξ) H n+1, and replace the operator matrix X M n (X ) with the matrix X 0 M n+1 (X ). The corresponding operators a = ɛ 2 I + n i=1 a i a i and b = ɛ 2 I + n i=1 b i b i are now invertible and the proof proceeds as before. Theorem 2.3 and 2.4 have the following consequence, which was first established by R. R. Smith in [47]. Theorem 2.5 (R. R. Smith). Let H be a Hilbert space, A B(H) be a C*- subalgebra with a cyclic vector and X B(H) be an A-bimodule. Suppose that Φ : A B(H) is an A-bimodular bounded linear map. Then Φ is completely bounded and Φ cb = Φ Harmonic analysis. Throughout these notes, will denote a locally compact group. For technical simplicity, we will assume throughout that is second countable. If E, F, we let as usual E 1 = {s 1 : s E}, EF = {st : s E, t F } and E n = {s 1 s n : s i E, i = 1,..., n} (n N). Left Haar measure on will be denoted by m, and it will be assumed to have total mass 1 if is compact. Integration along m with respect to the variable s will be written ds. We write L p () for the corresponding Lebesgue space, for 1 p, and M() for the space of all regular bounded Borel measures on. The Riesz Representation Theorem identifies M() with the Banach space dual of the space C 0 () of all continuous functions on vanishing at infinity; the duality here is given by f, µ = f(s)dµ(s), f C 0 (), µ M(). Note that M() is an involutive Banach algebra with respect to the convolution product defined through the relation f, µ ν = f(st)dµ(s)dν(t), f C 0 (), µ, ν M(),

5 and the involution given by HERZ-SCHUR MULTIPLIERS 5 µ (E) = µ(e 1 ), µ M(), E a Borel subset of. The space L 1 () can, by virtue of the Radon-Nikodym Theorem, be regarded as the closed ideal of all absolutely continuous with respect to m measures in M(). Note that the inherited convolution product on L 1 () turns it into an approximately unital involutive Banach algebra. The involution of L 1 () is given by f (s) = (s) 1 f(s 1 ), s, f L 1 (). Here, and in the sequel, denotes the modular function of, defined by the property m(es) = (s)m(e), s, E a Borel subset of. iven a complex function f on, we let ˇf(s) = f(s 1 ), f(s) = f(s 1 ), s. If H is a Hilbert space, we denote by U(H) the group of all unitary operators acting on H. A unitary representation of is a homomorphism π : U(H), continuous in the strong (equivalently, the weak) operator topology. We often write H π = H to designate the dependence of H on π. iven such π, there exists a non-degenerate *-representation of L 1 () (which we will denote with the same symbol) such that π(f) = f(s)π(s)ds, f L 1 (), in the norm topology of B(H π ). Two unitary representations π 1, π 2 of are called equivalent if there exists a unitary operator U B(H π1, H π2 ) such that Uπ 1 (s)u = π 2 (s), s. The set of all equivalence classes of irreducible unitary representations of is denoted by Ĝ and called the spectrum of. We think of Ĝ as a complete family of inequivalent irreducible representations of. A coefficient of π is a function on of the form s (π(s)ξ, η), where ξ, η H. The Fourier-Stieltjes algebra of is the collection of all coefficients of unitary representations of ; it is clear that B() is contained in the algebra C b () of all bounded continuous functions on. It is not difficult to see that B() is an algebra with respect to pointwise addition and multiplication. It is moreover a Banach algebra with respect to the norm u = inf{ ξ η : u( ) = (π( )ξ, η)}, where the infimum is taken over all unitary representations π and all vectors ξ and η with the designated property. For s, let λ s U(L 2 ()) be given by λ s f(t) = f(s 1 t), t, f L 2 (). The map λ : U(L 2 ()) sending s to λ s is a representation of, called the left regular representation. The corresponding representation of L 1 () is faithful (and non-degenerate). The Fourier algebra A() of

6 6 IVAN. TODOROV is the collection of all coefficients of λ; it is a closed ideal of B() and the norm on A() is given by u = inf{ ξ η : u(s) = (λ s ξ, η), s, ξ, η L 2 ()}. Through the pivotal work of P. Eymard [10], A() is a (commutative) semisimple regular Banach algebra with spectrum. We note that A() C 0 () and u u for every u A(). Moreover, B() C c () A() (here C c () stands for the space of all continuous functions on with compact support). We denote by C () the C*-algebra of ; this is the enveloping C*-algebra of L 1 (), that is, the completion of L 1 () with respect to the norm f = sup{ π(f) : π a unitary representation of } (note that the supremum on the right hand side is finite since, for every representation π of and every f L 1 (), we have π(f) f 1 ). The C*-algebra C () is characterised by the following universal property: for every unitary representation π of, there exists a unique non-degenerate representation π of C () such that π(f) = π(f) for every f L 1 (). In the future, we will not use a different notation for π and simply denote it by π. Note that L 1 () can be considered in a natural fashion as a norm dense *-subalgebra of C (). The reduced C*-algebra Cr () of is the closure, in the operator norm, of the image λ(l 1 ()) inside B(L 2 ()), while the group von Neumann algebra of is the weak* (equivalently, the weak, or the strong, operator topology) closure of Cr (). The Banach space dual of C () can be isometrically identified with B() via the formula (3) f, u = f(s)u(s)ds, f L 1 (), u B(). In a similar fashion, the (unique) Banach space predual of VN() can be isometrically identified with A(); in addition to the formula (3) (where u is taken from A()), the duality is described by the formulas u, λ s = u(s), s, u A(). For a given u A() and T VN(), the functional on A() given by v uv, T, v A(), is bounded and thus there exists a (unique) element u T VN() such that v, u T = uv, T, Note that u T u T. The map v A(). A() VN() VN(), (u, T ) u T, is easily seen to define the structure of a Banach A()-module on VN(). It can be shown that in fact VN() is an operator A()-module when equipped with this action; moreover, for each u A(), the map T u T is weak* continuous.

7 HERZ-SCHUR MULTIPLIERS 7 The positive linear functionals on C () correspond to positive definite functions from B(). A function u C() (where C() is the space of all continuous functions on ) is called positive definite if the matrix (u(s i s 1 j )) n i,j=1 is positive for every choice s 1,..., s n of elements of. Equivalently, u is positive definite if, viewed as an element of L (), it defines a positive linear functional on L 1 (), that is, if u(s)(f f )(s)ds 0, f L 1 (). We denote by P () the collection of all continuous positive definite functions on ; it is easy to see that P () C b () and that if u P () then u = u(e). Using NS theory, one can show that P () B(). More precisely, a function u B() is positive definite if and only if there exists a representation π of and a vector ξ H π (cyclic for π) such that u(s) = (π(s)ξ, ξ), s. We then say that u is a positive coefficient of π. Let N λ be the kernel of the left regular representation λ of C (). Clearly, Cr () = C () /N λ, up to a *-isomorphism. The group is called amenable if N λ = {0}; in this case, the C*-algebras C () and Cr () are *-isomorphic. We note that amenability is usually defined by requiring the existence of a left invariant mean on L (). A further equivalent formulation of amenability can be derived as follows. iven two families S and T of representations of (or, equivalently, of C ()), say that S is weakly contained in T if π T ker π π S ker π. Every positive linear functional on Cr () gives rise, via composition with the corresponding quotient map, to a positive linear functional on C () and can hence be identified with an element of P (). It was shown by J. M.. Fell [11] that the elements of P () obtained in this way are precisely the positive coefficients of the unitary representations of weakly contained in λ. More precisely, we have the following facts. Proposition 2.6. Let u P (). The following are equivalent: (i) The formula λ(f) u(s)f(s)ds, f L 1 (), defines a positive linear functional on Cr (); (ii) The NS representation corresponding to u is weakly contained in λ; (iii) The function u is the limit, in the topology of uniform convergence on compacts, of functions of the form f f, f L 2 (). We note that the functions of the form f f are precisely the positive coefficients of the left regular representation of : (λ s (f), f) = f(s 1 t)f(t)dt = (f f)(s), s.

8 8 IVAN. TODOROV The set of functions satisfying the equivalent conditions of Proposition 2.6 will be denoted by P λ (), and its linear span in B(), by B λ (). Thus, the space B λ () corresponds in a canonical fashion to the Banach space dual of C r (). We have the following additional characterisation of B λ (): Proposition 2.7. The following are equivalent, for a function u : C: (i) u B λ (); (ii) the formula λ(f) u(s)f(s)ds, f L 1 (), defines a bounded linear functional on C r (); From Proposition 2.7 one can easily derive that the group is amenable if and only if the constant function 1 can be approximated, uniformly on compact sets, by elements of A(). Equivalently, is amenable if and only if A() possesses a bounded approximate identity [34]. Since B() = C (), we can equip B() with the operator space structure arising from this duality. Similarly, A() (resp. B λ ()), being the predual of VN() (resp. the dual of C r ()), can be equipped with a canonical operator space structure. Throughout these notes, any reference to A(), B() and B λ () as operator spaces utilises the structures just introduced. If and H are locally compact groups, we denote as customary by H the direct product of and H equipped with the product topology. We have that VN( H) = VN() VN(H). It follows that, up to a complete isometry, A( H) = A() ˆ A(H), where ˆ denotes the operator projective tensor product. 3. The spaces MA() and M cb A() In this section, we define Herz-Schur multipliers and establish some of their basic properties. We follow closely [5], where Herz-Schur multipliers were first introduced and studied. Definition 3.1. A function u : C is called a multiplier of A() if uv A() for every v A(). We denote the set of all multipliers of A() by MA(). Clearly, MA() is an algebra with respect to pointwise addition and multiplication. We note that if u MA() then u is continuous; indeed, given s, choose a compact neighbourhood K of s and let v A() be a function with v K = 1. Then uv K = u K, and since uv is continuous, it follows that u is continuous at s. If u MA(), let m u : A() A() be the map given by m u (v) = uv, v A(). We note that the map m u satisfies the relation m u (vw) = vm u (w) for all v, w A(). Proposition 3.2. If u MA() then the map m u is bounded.

9 HERZ-SCHUR MULTIPLIERS 9 Proof. The (linear) map m u is defined on a Banach space; in order to show that m u is bounded, it suffices, by the Closed raph Theorem, to show that m u has closed graph. Let therefore (u k ) k N A() be a null sequence such that uu k v for some v A(). Then u k k 0 and uu k v k 0. Thus, v(s) = lim k u(s)u k(s) = 0, s ; in other words, v = 0 as an element of A(). Exercise 3.3. Suppose that T : A() A() is a linear map such that T (vw) = vt (w), v, w A(). Then there exists u MA() such that T = m u. For u MA(), we set u m def = m u. Remark 3.4. We have that B() MA(). Moreover, if u B() then u m u B(). Proof. Since A() is an ideal of B(), we have that B() MA(). Moreover, m u = sup{ uv : v A(), v 1} u B(). Definition 3.5. An element v MA() is called a completely bounded multiplier of A() if the map m v is completely bounded. Let M cb A() be set of all completely bounded multipliers of A(). Since m uv = m u m v (for u, v MA()), we have that M cb A() is a subalgebra of MA(). Set u cbm = m u cb (where u M cb A()); then M cb A() is a Banach algebra with respect to cbm. If u MA(), the dual map m u of m u acts on VN(); we will denote it by S u. If s and v A() then v, S u (λ s ) = m u (v), λ s = uv, λ s = u(s)v(s) = v, u(s)λ s. This shows that S u (λ s ) = u(s)λ s, s. In particular, it follows that (4) v(s) = v(s)λ s = S v (λ s ) v m, s ; and thus the elements of M A() are bounded functions. The above argument also proves a part of the following theorem. Theorem 3.6. Let u : C be a bounded continuous function. The following are equivalent: (i) u MA(); (ii) There exists a (unique) bounded weak* continuous linear map T on VN() such that T (λ s ) = u(s)λ s, s ; (iii) There exists a bounded linear map R on Cr () such that R(λ(f)) = λ(uf), (iv) uv B λ () for every v B λ (). f L 1 ();

10 10 IVAN. TODOROV Proof. (i) (ii) follows from the argument before the statement of the theorem, by taking T = S u. (ii) (iii) We claim that the restriction R of T to Cr () satisfies the given relations. Indeed, letting f L 1 (), we have that, in the topology of the norm, λ(f) = f(s)λ sds. Since T is norm continuous, (5) T (λ(f)) = f(s)t (λ s )ds = f(s)u(s)λ s ds. Since u is bounded (see (4)), uf L 1 () and (5) shows that T (λ(f)) = λ(uf). (iii) (iv) For v B λ () and f L 1 (), we have u(s)v(s)f(s)ds = λ(uf), v R λ(f) v B λ (). It follows that the map λ(f) u(s)v(s)f(s)ds, f L 1 (), extends to a bounded linear functional on Cr () of norm not exceeding R v Bλ (). By Proposition 2.7, uv B λ (). (iv) (i) An application of the Closed raph Theorem as in the proof of Proposition 3.2 shows that the map v uv on B λ () is bounded. Suppose that v B() C c (); then uv B λ () C c () A(). Since B() C c () = A() [10], it follows that ua() A(). Remark 3.7. Let u L (). The following are equivalent: (i) u is equivalent (with respect to the Haar measure) to a function from MA(); (ii) there exists C > 0 such that λ(uf) C λ(f), f L 1 (). Proof. (i) (ii) follows from Theorem 3.6 and the fact that if u v then λ(uf) = λ(vf) for every f L 1 (). (ii) (i) Let v B λ () and ω : λ(l 1 ()) C be the functional given by ω(λ(f)) = ufvdm, f L 1 (). Then ω(λ(f)) C v Bλ () λ(f), Thus, there exists w B λ () such that ω(λ(f)) = wfdm, f L 1 (). f L 1 (). It follows that uv = w almost everywhere. Since such a function w exists for every choice of v B λ (), we conclude that u agrees almost everywhere with a continuous function. The statement in (i) now follows from Theorem 3.6.

11 HERZ-SCHUR MULTIPLIERS 11 Note that u M cb A() if and only if the map T, or that map R, from Theorem 3.6 are in fact completely bounded. We next characterise the elements of M cb A() within MA(). If H is another locally compact group and u : C, we write u 1 for the function defined on H by u 1(s, t) = u(s), s, t H. To underline the dependence of this function on H, we write u 1 H. Recall that SU(n) denotes the special unitary group in dimension n, that is, the group (under multiplication) of all n by n unitary matrices with determinant 1. Theorem 3.8. Let u MA(). The following are equivalent: (i) u M cb A(); (ii) u 1 MA( H) for every locally compact group H; (iii) u 1 MA( SU(2)). Moreover, if these conditions are fulfilled then u cbm = sup u 1 H m = u 1 SU(2) m. H l.c.g. Proof. (i) (ii) By assumption, the map S u : VN() VN() is completely bounded and weak* continuous. Let H = L 2 (H). The map S u id : VN() B(H) VN() B(H) is bounded and weak* continuous with S u id S u cb. We have that (S u id)(t S) = S u (T ) S, T VN(), S B(H). In particular, if s and t H then (S u id)(λ s λ t ) = S u (λ s ) λ t = (u 1)(s, t)λ (s,t). By Theorem 3.6, u 1 MA( H). (ii) (iii) is trivial. (iii) (i) The group SU(2) is compact; by the Peter-Weyl Theorem, VN(SU(2)) = l π SU(2) ˆ B(H π) as von Neumann algebras. It is well-known that for every n N there exists a unique equivalence class of irreducible unitary representations of SU(2) whose underlying Hilbert space has dimension n. Thus, VN(SU(2)) = n=1 M n. It follows that (6) VN() VN(SU(2)) = n=1 VN() M n. For s and t SU(2), we have that S u 1 (λ s λ t ) = S u (λ s ) λ t. By linearity and weak* continuity, we have that S u 1 = S u id SU(2). By (6), if T = n=1 T n VN() VN(SU(2)) then S u 1 (T ) = n=1 S(n) u (T n ). Thus, S u (n) (T n ) u 1 m T n, n N. Since {T n : T VN() VN(SU(2))} = VN() M n, we conclude that S u (n) u 1 m, and (i) is established. Corollary 3.9. We have B() M cb A(). Moreover, if u B() then u cbm u B(). Proof. Let u B() and H be any locally compact group. Then u 1 H B( H); indeed, if π : B(H π ) is a unitary representation of then π 1 : H B(H π ) given by π 1(s, t) = π(s) is a unitary

12 12 IVAN. TODOROV representation of H. It follows that u 1 H B( H); moreover, u 1 H B( H) u B(). By Remark 3.4, u 1 H MA( H) and u 1 H m u 1 H B( H) u B(). It follows by Theorem 3.8 that u M cb A() and u cbm u B(). It follows from Corollary 3.9 that B() M cb A() MA(). It was shown by V. Losert [35] that is amenable if and only if B() = MA(). The following simple observation will be useful in the sequel. Proposition If u A() then S u (T ) = u T for every T VN(). Proof. If s and v A() then S u (λ s ), v = u(s)λ s, v = u(s)v(s)λ s = λ s, uv = u λ s, v. The claim follows by linearity and weak* continuity The case of commutative groups. In this subsection, we follow the exposition of [44]. We assume throughout that is abelian. We briefly recall some basic facts about Fourier theory on. Let Γ = Ĝ be the dual group of. If f L 1 (Γ), let ˆf : C be its Fourier transform, namely, the function ˆf(s) = f(γ)γ(s)dγ, s. We also set F(f) = ˆf, f L 1 (Γ). Then Γ F(f) 2 = f 2, f L 1 (Γ) L 2 (Γ), and thus F extends to an isometry (denoted again by F) from L 2 (Γ) onto L 2 (). We often write ˆf = F(f) for elements f of L 2 (Γ). Note that, if f, g L 1 (Γ), then F(f g) = F(f)F(g). This implies that if f, g L 2 (Γ) are such that ˆf, ĝ L 1 () then fg = ˆf ĝ. These observations form the base for the following fact. Proposition We have that A() = { ˆf : f L 1 (Γ)}. Moreover, the map f ˆf is an isometric homomorphism of L 1 (Γ) onto A(). Fourier transform gives a useful insight into the C*-algebra and the von Neumann algebra of Γ. Indeed, let L () act on L 2 () via multiplication; more precisely, consider the algebra D = {M ϕ : ϕ L ()}, where M ϕ B(L 2 ()) is given by M ϕ f = ϕf, f L 2 (). Let also C = {M ϕ : ϕ C 0 ()}, A straightforward calculation shows that (7) Fλ(f)F = M ˆf, f L 1 (Γ).

13 It follows that HERZ-SCHUR MULTIPLIERS 13 F VN(Γ)F = D, FC r (Γ)F = C. Our next aim is to characterise similarly B() and to show that MA() coincides with it. First note that the Fourier transform can be extended to the algebra M() of all Radon measures on Γ; for µ M(), set ˆµ(s) = γ(s)dµ(γ), s. Γ Note that ˆµ is a continuous function on with ˆµ µ (the latter norm being the total variation of µ). The following is a classical result of Bochner s. Theorem We have that Thus, B() = {ˆµ : µ M(Γ)}. P () = {ˆµ : µ M(Γ), positive}. Proof. Let u P (), and assume, without loss of generality, that u(e) = 1. Via the identification of B() with C (), the function u corresponds to a state ω u of C (). The Cauchy-Schwarz inequality for positive linear functionals now implies that (8) ω u (f) 2 ω u (f f), f L 1 (). Fix f L 1 () and let h = f f. Then a successive application of (8) shows that ω u (f) 2 ( h 2n 1 ) 2 n. Taking a limit, we obtain that ω u (f) 2 r(h), where r(h) is the spectral radius of h as an element of the Banach algebra L 1 (). We have that r(h) = ĥ, and hence ω u (f) 2 ĥ. It now follows that the map ˆf ω u (f) is well-defined and bounded in the uniform norm. On the other hand, an application of the Stone-Weierstrass Theorem shows that A(Γ) is dense in C 0 (Γ) in. By the Riesz Representation Theorem, there exists a positive measure µ M(Γ) such that ω u (f) = ˆfdµ, f L 1 (). Γ Thus, f(s)u(s)ds = ω u (f) = f(s)γ(s)dµ(γ)ds, f L 1 (). Γ It now follows that u = ˆµ almost everywhere. Since both u and ˆµ are continuous, we conclude that u = ˆµ everywhere.

14 14 IVAN. TODOROV Conversely, suppose that µ M(Γ) is a positive measure. For any choice s 1,..., s n of points in, and any choice of scalars λ 1,..., λ n, we have n n λ i λ j ˆµ(s i s j ) = λ i λ j γ(s i s j )dµ(γ) i,j=1 = Γ i,j=1 n λ i γ(s i )λ j γ(s j )dµ(γ). Γ i,j=1 Since (γ(s i )γ(s j )) n i,j=1 is a positive matrix for all γ Γ, we have that n λ i γ(s i )λ j γ(s j ) 0 i,j=1 for all γ Γ. Since µ is positive, we conclude that n λ i λ j ˆµ(s i s j ) 0. i,j=1 This shows that ˆµ is a positive definite function. The second equality follows from the fact that B() is the linear span of P (). Remark It can be shown that, if µ M(Γ), then ˆµ B() = µ, where the latter denotes the total variation of µ. We have the following alternative description of B(). Let T (Γ) be the linear space of all trigonometric polynomials on Γ, that is, the space of all functions f : Γ C of the form n (9) f(γ) = c i γ, s i, γ Γ, i=1 where s i and c i C, i = 1,..., n. We note that, equivalently, a trigonometric polynomial of the form (9) can be identified with the element T f VN() given by n T f = c i λ si. i=1 Proposition Let u : C be a continuous function. The following are equivalent: (i) u B() and u C; (ii) if f is a trigonometric polynomial of the form (9) then n (10) c i u(s i ) C f. i=1

15 HERZ-SCHUR MULTIPLIERS 15 Proof. (ii) (i) Let d be the group equipped with the discrete topology. For f T (Γ), we have, in view of (7), that T f = f. Thus, (10) implies that the linear map ω : T f n i=1 c iu(s i ) has the property ω(t f ) C T f, f T (Γ). Since {T f : f T (Γ)} is dense in C ( d ) in norm, the functional ω has an extension to a bounded linear functional on C ( d ). Thus, there exists v B( d ) such that n ω(t f ) = T f, v = c i v(s i ), f T (Γ). i=1 It follows that u = v. However, by the Bochner-Eberlein Theorem, B( d ) C() = B(), and the proof is complete. (i) (ii) By virtue of the Bochner-Eberlein Theorem, B( d ) C() = B(), and hence u B( d ). The claim now follows from the fact that, if f is as in (9), then T f, u = n i=1 c iu(s i ). Theorem Suppose that u : C is a function such that uv B() for every v A(). Then u B(). In particular, MA() = B(). Moreover, if u B() then u m = u. Proof. One can easily show that u is continuous; moreover, a straightforward application of the Closed raph Theorem (see the proof of Proposition 3.2) shows that the map T : A() B() given by T (v) = uv, is bounded. Let s 1,..., s n, c 1,..., c n C, and f = n i=1 c is i T (Γ) be the corresponding trigonometric polynomial on Γ. For a given ɛ > 0, let v A() be a function such that v(s i ) = 1, i = 1,..., n, and v 1 + ɛ. Then uv(s i ) = u(s i ), i = 1,..., n. Since uv B(), Theorem 3.12 gives an element µ M(Γ) such that ˆµ = uv. Thus, n n c i u(s i ) = c iˆµ(s i ) i=1 i=1 ( n ) = c i γ(s i ) dµ(γ) µ f. Γ i=1 By Proposition 3.13, u B() and u µ = uv B() T (1 + ɛ); thus, u T. We have that T = u m since the image of the map T is in A(). Thus, u u m ; by Corollary 3.9, we have that u = u m. Corollary Let be a locally compact abelian group. Then M cb A() = B(). Moreover, if u B() then u = u cbm. 4. Schur multipliers This section is dedicated to a brief introduction to measurable Schur multipliers, which will be used in subsequent parts of the present text.

16 16 IVAN. TODOROV 4.1. ω-topology. We fix for the whole section standard measure spaces (X, µ) and (Y, ν); by this we mean that there exist locally compact, metrisable, complete topologies on X and Y (called the underlying topologies), with respect to which µ and ν are regular Borel σ-finite measures. By a measurable rectangle we will mean a subset of X Y of the form α β, where α and β are measurable. We denote by µ ν the product measure (defined on the product σ-algebra on X Y, that is, on the σ- algebra generated by all measurable rectangles). A subset E X Y will be called marginally null if there exist null sets M X and N Y such that E (M Y ) (X N). Every marginally null subset of X Y is clearly a µ ν-null set. The converse is not true; for an example, consider the subset = {(x, x) : x [0, 1]} of [0, 1] [0, 1], where the unit interval [0, 1] is equipped with Lebesgue measure. Two measurable sets E, F X Y will be called marginally equivalent if the symmetric difference of E and F is marginally null; in this case we write E = F. The sets E and F will be called equivalent if their symmetric difference is µ ν-null; in this case we write E F. Similarly, for measurable functions ϕ, ψ : X Y C, we write ϕ ψ (resp. ϕ = ψ) if the set {(x, y) : ϕ(x, y) ψ(x, y)} is null (resp. marginally null). A measurable subset κ X Y is called ω-open if it is marginally equivalent to a subset of X Y of the form i=1 α i β i, where α i X and β i Y are measurable, i N. The set κ will be called ω-closed if its complement κ c is ω-open. The set of all ω-open sets is a pseudo-topology, that is, it is closed under taking countable unions and finite intersections. Lemma 4.1 ([9]). Suppose that the underlying topologies of X and Y are compact and the measures µ and ν are finite. Let κ be an ω-closed set, and γ k, k N, be ω-open subsets, of X Y, such that κ k=1 γ k. For every ɛ > 0 there exist measurable sets X ɛ X and Y ɛ Y such that µ(x \ X ɛ ) < ɛ, µ(y \ Y ɛ ) < ɛ and the set κ (X ɛ Y ɛ ) is contained in the union of finitely many of the sets γ k, k N. A function h : X Y C will be called ω-continuous if h 1 (U) is ω- open for every open set U C. Let C ω (X Y ) be the set of all (marginal equivalence classes of) ω-continuous functions on X Y. The following facts will be useful; their proofs are left as an exercise. Proposition 4.2. (i) The set C ω (X Y ) is an algebra with respect to pointwise addition and multiplication. (ii) If ϕ, ψ C ω (X Y ) and ϕ ψ then ϕ = ψ The predual of B(H 1, H 2 ). We let H 1 = L 2 (X, µ) and H 2 = L 2 (Y, ν). It is well-known that the dual Banach space of the space C 1 (H 2, H 1 ) of all trace class operators from H 2 into H 1 is isometrically isomorphic to B(H 1, H 2 ), the duality being given by S, T = tr(st ), S C 1 (H 2, H 1 ), T B(H 1, H 2 ),

17 HERZ-SCHUR MULTIPLIERS 17 where tr denotes the canonical trace on C 1 (H 1 ). In this subsection we describe an identification of C 1 (H 2, H 1 ) with a certain function space on X Y, which will be used in the rest of the section. Recall first that C 1 (H 2, H 1 ) can be naturally identified with the projective tensor product H 1 ˆ H 2 by identifying an elementary tensor f g, where f H 1 and g H 2, with the operator T f g of rank one given by ( ) T f g (h) = (h, g)f = h(y)g(y)dν(y) f, h H 2. Y In this way, the operators of finite rank from H 2 into H 1 are identified with elements of the algebraic tensor product L 2 (X, µ) L 2 (Y, ν. Lemma 4.3. Suppose that n j=1 f j g j = 0 as an element of L 2 (X, µ) L 2 (Y, ν). Then n j=1 f j(x)g j (y) = 0 for marginally almost all (x, y). Proof. Let ψ(x, y) = n j=1 f j(x)g j (y), (x, y) X Y. The function ψ is well-defined up to a marginally null set. We first note that Reψ arises from the element 1 n f j g j + 1 n f j g j 2 2 j=1 of L 2 (X, µ) L 2 (Y, ν), which coincides with the zero element since both terms are zero. Similarly, Imψ arises from the element 1 n f j g j 1 n f j g j 2i 2i j=1 of L 2 (X, µ) L 2 (Y, ν) which is zero. If we show that Reψ and Imψ, viewed as functions, are equal to zero marginally almost everywhere, the lemma will be established. We may hence assume that the function ψ takes real values. By Proposition 4.2, ψ is ω-continuous. Suppose that ψ is not marginally equivalent to the zero function; without loss of generality, assume that there exist δ > 0 and a rectangle α β of finite non-zero measure such that ψ(x, y) > δ for all (x, y) α β. But then n 0 < ψ(x, y)dµ(x)dν(y) = (f j, χ α )(g j, χ β ) = 0, a contradiction. α β For an element u = n j=1 f j g j L 2 (X, µ) L 2 (Y, ν), we let ψ u be the function on X Y given by ψ u (x, y) = n j=1 f j(x)g j (y). By Lemma 4.3, ψ u is well-defined, as an element of C ω (X Y ). Lemma 4.4. Let {u n } n=1 L2 (X, µ) L 2 (Y, ν) be a sequence converging to zero in the projective tensor norm, and ψ n = ψ un. Then there exists a subsequence {n k } k=1 of natural numbers such that ψ n k k 0 marginally almost everywhere. j=1 j=1 j=1

18 18 IVAN. TODOROV Proof. We may assume that u n = p n j=1 f (n) j Thus, p n j=1 f (n) j 2 2 n 0, X p n f (n) j=1 p n j=1 g (n) j, and g (n) j 2 2 n 0. j (x) 2 dµ(x) n 0 and hence there exists a subsequence {n k } k=1 of natural numbers such that p nk j=1 We may assume that, moreover, p nk j=1 By the Cauchy-Schwarz inequality, f (n k) j (x) 2 n 0 almost everywhere. g (n k) j (y) 2 n 0 almost everywhere. p nk ψ nk (x, y) 2 j=1 marginally almost everywhere. p f (n nk k) j (x) 2 j=1 g (n k) j (y) 2 k 0 Now let u L 2 (X, µ) ˆ L 2 (Y, ν), and suppose that u = j=1 f j g j, where j=1 f j 2 2 < and j=1 g j 2 2 <. Since j=1 f j 2 2 < we have that f j (x) 2 < almost everywhere on X and j=1 g j (x) 2 < almost everywhere on Y. j=1 By the Cauchy-Schwarz inequality, the sum j=1 f j(x)g j (y) is finite for marginally all (x, y). Let ψ = ψ u be the complex function defined marginally almost everywhere on X Y by letting (11) ψ(x, y) = f j (x)g j (y). j=1 We note that the function ψ(x, y) does not depend on the representation of u. To this end, suppose that u = j=1 ξ j η j is another representation of u and let φ(x, y) = j=1 ξ j(x)η j (y). Set u n = n j=1 f j g j, v n = n j=1 ξ j η j, ψ n (x, y) = n j=1 f j(x)g j (y) and φ n (x, y) = n j=1 ξ j(x)η j (y).

19 HERZ-SCHUR MULTIPLIERS 19 Thus, ψ n (x, y) ψ(x, y) and φ n (x, y) φ(x, y) marginally almost everywhere. We have that u n v n n 0; by Lemma 4.4, there exists a subsequence {n k } k=1 of natural numbers such that ψ n k (x, y) φ nk (x, y) 0 marginally almost everywhere. Thus, ψ(x, y) = φ(x, y) marginally almost everywhere. We now let T (X, Y ) be the space of all classes (with respect to marginal equivalence) of functions ψ u, associated to elements u L 2 (X, µ) ˆ L 2 (Y, ν). def We equip T (X, Y ) with the norm ψ u = u. It is easy to note that, conversely, if ψ : X Y C is a function which admits a representation of the form (11), where j=1 f j 2 2 < and j=1 g j 2 2 <, then ψ = ψ u, where u = i=1 f i g i. If u = i=1 f i g i, let T u : H 2 H 1 be the nuclear operator given by T u (η)(x) = (η, g i )f i, η H 2. i=1 It is immediate that T u is an integral operator with integral kernel ψ u. We note that if k L 2 (Y X), T k C 2 (H 1, H 2 ) is the corresponding Hilbert-Schmidt operator given by T k ξ(y) = k(y, x)ξ(x)dµ(x), y Y, and if u T (X, Y ) then (12) T u, T k = X X Y ψ u (x, y)k(y, x)dµ ν(x, y). Indeed, (12) can be verified first in the case T ψ is an operator of rank one and then its validity follows by linearity and weak* continuity. If u T (X, Y ) and T B(H 1, H 2 ), we will often write u, T for T u, T. Remark 4.5. The map sending an element u of L 2 (X, µ) ˆ L 2 (Y, ν) to its corresponding class (with respect to marginal equivalence) of functions in T (X, Y ) is injective. That is, if u 1, u 2 L 2 (X, µ) ˆ L 2 (Y, ν) and ψ u1 = ψu2 then T u1 = T u2. Proof. This is immediate from the fact that T u1 and T u2 are integral operators with integral kernels ψ u1 and ψ u2, respectively. Henceforth, we identify the space of (marginal equivalence classes of) functions T (X, Y ) with the projective tensor product L 2 (X, µ) ˆ L 2 (Y, ν); we thus suppress the distinction between u and ψ u and use the same symbol to denote them. We note that equation (12) implies the following, which will be useful in the sequel: suppose that ψ T (X, Y ) and ψ is a measurable function with ψ ψ. Then, clearly, ψ(x, y)k(y, x)dµ ν(x, y) = ψ (x, y)k(y, x)dµ ν(x, y), X Y X Y

20 20 IVAN. TODOROV for all k L 2 (Y X). It follows that the map T k ψ (x, y)k(y, x)dµ ν(x, y) X Y is bounded in the operator norm, and hence there exists ψ T (X, Y ) such that T ψ, T k = ψ (x, y)k(y, x)dµ ν(x, y). X Y It now follows that ψ ψ, and thus ψ ψ. Since both ψ and ψ are ω- continuous, we have by Proposition 4.2 that ψ = ψ. Thus, the function ψ is the integral kernel of the operator T ψ. In other words, integral operators T ψ can be defined unambiguously for any function ψ that is equivalent, with respect to the product measure, to a function from T (X, Y ). Proposition 4.6. The inclusion T (X, Y ) C ω (X Y ) holds. Proof. We first establish the following Claim. If f n : X Y C and φ n : X Y R + are ω-continuous functions such that inf n φ n (x, y) = 0 for marginally almost all x, y, and if f : X Y C is a function with f(x, y) f n (x, y) φ n (x, y) for marginally almost all x, y, then f is ω-continuous. Proof of Claim. It is easy to reduce the statement to the case where f n and f are real valued. In this case, however, for any a R, up to a marginally null set, f 1 ((a, + )) = fn 1 ((a + 1 m, + )) φ n((0, 1 m )). The claim now follows. m,n=1 Let h = i=1 f i g i, where i=1 f i 2 2 < and i=1 g i 2 2 <. Set φ n+1 (x, y) = i=n ( f i(x) 2 + g i (y) 2 ), n N. Then the functions φ n are ω-continuous, inf n φ n (x, y) = 0 for marginally almost all x, y and if we let h n = n i=1 f i g i we see that h n h φ n up to a marginally null set, for each n. The statement is now immediate by the Claim The space T (). Let be a locally compact group. We write T () = T (, ). The map P : T () A(), given by (13) P (f g)(t) = λ t, f g = (λ t f, g) = f(t 1 s)g(s)ds = g ˇf(t) is a contractive surjection, by the definition of A(). The next lemma will be useful later. Lemma 4.7. If h T () then (14) P (h)(t) = h(t 1 s, s)ds, t.

21 HERZ-SCHUR MULTIPLIERS 21 Proof. Identity (14) is a direct consequence of (13) if h is a finite sum of elementary tensors. Let h = i=1 f i g i T (), where i=1 f i 2 2 < and i=1 g i 2 2 <, and let h n be the nth partial sum of this series. By the continuity of P, P (h n ) P (h) 0 in A(); since is dominated by the norm of A(), we conclude that P (h n )(t) P (h)(t) for every t. By Lemma 4.4, there exists a subsequence (h nk ) k N of (h n ) n N such that h nk h marginally almost everywhere. It follows that, for every t, one has h nk (t 1 s, s) h(t 1 s, s) for almost all s. By [36, (4.3)], the function s i=1 f i(t 1 s) g i (s) is integrable, and hence an application of the Lebesgue Dominated Convergence Theorem shows that h n k (t 1 s, s)ds k h(t 1 s, s)ds, for every t. The proof is complete The characterisation theorem. If h : X Y C is a function then, by writing h µ ν T (X, Y ), we will mean that h is equivalent, with respect to the measure µ ν, to a function that lies in T (X, Y ). If h µ ν T (X, Y ) then there exists a unique, up to marginal equivalence, element h of T (X, Y ) such that h h. Indeed, if h h and h h, where h, h T (X, Y ), then h h and, by Propositions 4.6 and 4.2, h = h. Definition 4.8. A function ϕ L (X Y ) is called a Schur multiplier if ϕh µ ν T (X, Y ) for every h T (X, Y ). Let S µ,ν (X, Y ) be the set of all Schur multipliers on X Y with respect to a pair of fixed measures µ, ν. When the measures are understood from the context, we simply write S(X, Y ). We note that, strictly speaking, Schur multipliers are classes of functions with respect to almost everywhere equality. If ϕ S(X, Y ), let m ϕ : T (X, Y ) T (X, Y ) be given by m ϕ h = ϕh. Note that, strictly speaking, m ϕ h is (defined to be) the (unique, up to marginal equivalence) function h T (X, Y ) such that h ϕh. We also note that if ϕ, ψ S(X, Y ) and ϕ ψ then m ϕ = m ψ. Thus, the map m ϕ is independent of the representative ϕ we use to define it. Proposition 4.9. If ϕ S(X, Y ) then the operator m ϕ on T (X, Y ) is bounded. Proof. We apply the Closed raph Theorem. Suppose that (h k ) k N T (X, Y ) is such that h k 0 and ϕh k h 0 for some h T (X, Y ). Let h k be the unique element from T (X, Y ) such that ϕh k h k, k N. Using Lemma 4.4, we may assume, after passing to subsequences, that h k 0 and h k h marginally almost everywhere. It follows that ϕh k h almost everywhere, and hence h = 0 almost everywhere. Since h is ω-continuous, Proposition 4.2 implies that h = 0 marginally almost everywhere, and thus h = 0 as an element of T (X, Y ). If ϕ S(X, Y ), we write ϕ S = m ϕ. Our next aim is to give a characterisation of Schur multipliers; we follow the approach of [29]. For

22 22 IVAN. TODOROV ϕ S(X, Y ), we let S ϕ = m ϕ; thus, S ϕ : B(H 1, H 2 ) B(H 1, H 2 ) is a bounded weak* continuous map. For a L (X, µ) let M a be the operator on L 2 (X, µ) defined by M a f = af. Let D X = {M a : a L (X, µ)}; define D Y analogously. For a function ϕ : X Y C, let ˆϕ : Y X C be the function given by ˆϕ(y, x) = ϕ(x, y). Theorem Let ϕ S(X, Y ). Then S ϕ is a weak* continuous completely bounded D Y, D X -module map and, if k L 2 (Y X), then S ϕ (T k ) = T ˆϕk. Conversely, if Φ : B(H 1, H 2 ) B(H 1, H 2 ) is a a weak* continuous bounded D Y, D X -module map then there exists a unique ϕ S(X, Y ) such that Φ = S ϕ. Proof. Suppose that ϕ S(X, Y ). The fact that S ϕ is a bounded weak* continuous map was observed after the proof of Proposition 4.9. Let k L 2 (Y X) and h T (X, Y ). Using (12), we have S ϕ (T k ), T h = T k, m ϕ (T h ) = T k, T ϕh = k(y, x)ϕ(x, y)h(x, y)dµ ν(x, y) = T ˆϕk, T h. X Y Thus, S ϕ (T k ) = T ˆϕk. Now let a L (X, µ) and b L (Y, ν); for k L 2 (Y X) and h T (X, Y ) we have S ϕ (M b T k M a ), T h = a(x)b(y)ϕ(x, y)k(y, x)h(x, y)dµ ν(x, y) X Y = S ϕ (T k ), M a T h M b = M b S ϕ (T k )M a, T h ; thus, S ϕ is a D Y, D X -module map. It is easy to see that D X and D Y have cyclic vectors. By Theorem 2.5, S ϕ is completely bounded. The proof of the converse direction follows the lines of [29]. Suppose that Φ : B(H 1, H 2 ) B(H 1, H 2 ) is a a weak* continuous bounded D Y, D X - module map. By Theorem 2.5, Φ is completely bounded. By a well-known result of U. Haagerup s [17], there exists a bounded (row) operator B = (M bk ) k N M 1, (D Y ) and a bounded (column) operator A = (M ak ) k N M,1 (D X ) such that Φ(T ) = M bk T M ak, T B(H 1, H 2 ), k=1 where the series converges in the weak* topology. We have that C 1 = esssup x X k=1 a k (x) 2 < and C 2 = esssup y Y b k (y) 2 <, k=1

23 and hence the function HERZ-SCHUR MULTIPLIERS 23 ϕ(x, y) = a k (x)b k (y) k=1 is well-defined up to a marginally null set. We show that ϕ S(X, Y ); let h = i=1 f i g i T (X, Y ). Then However, a k f i 2 2 = k,i similarly, ϕ(x, y)h(x, y) = k,i X a k (x)f i (x)b k (y)g i (y), a k (x)f i (x) 2 dµ(x) C 1 k,i = C 1 f i 2 2; i=1 b k g i 2 2 C 2 g i 2 2, k,i i=1 X m.a.e.. f i (x) 2 dµ(x) and we are done. For k L 2 (Y X) we now have Φ(T k ) = S ϕ (T k ); since both Φ and S ϕ are bounded and weak*-continuous, we conclude by the weak* density of C 2 (H 1, H 2 ) in B(H 1, H 2 ) that Φ = S ϕ. Theorem 4.10 and its proof show the following. Corollary The map from S(X, Y ) into the space CB w D Y,D X (B(H 1, H 2 )) of all completely bounded weak* continuous D Y, D X -module maps, sending ϕ to S ϕ, is a bijective isometry. Exercise 4.12 ([48]). Show that the map from Corollary 4.11 is a complete isometry. In the sequel, we call ϕ the symbol of S ϕ and equip S(X, Y ) with the operator space structure that makes the map ϕ S ϕ a complete isometry. By a well-known result of U. Haagerup s [17] (see also [1]), the space CBD w Y,D X (B(H 1, H 2 )) is completely isometric and weak* homeomorphic to the weak* Haagerup tensor product D Y w h D X via the mapping sending an element k=1 B k A k D Y w h D X to the map T k=1 B kt A k. Utilising the canonical isomorphism between D X (resp. D Y ) and L (X, µ) (resp. L (Y, ν)), we see that D Y w hd X can be viewed as a space of (equivalence classes of) functions, and that it can be identified with S(X, Y ). We summarise this as a part of the theorem that follows. Theorem Let ϕ L (X Y ). The following are equivalent: (i) ϕ S(X, Y ) and ϕ S C; i

24 24 IVAN. TODOROV (ii) there exists sequences (a k ) k=1 L (X, µ) and (b k ) k=1 L (Y, ν) with def C 1 = esssup a k (x) 2 def C and C 2 = esssup b k (y) 2 C, such that x X k=1 ϕ(x, y) = y Y k=1 a k (x)b k (y) a.e. on X Y ; k=1 (iii) there exist a separable Hilbert space K and weakly measurable functions a : X K, b : Y K, such that esssup a(x) C, x X esssup b(y) C y Y and ϕ(x, y) = (a(x), b(y)), a.e. on X Y ; (iv) T ˆϕk C T k for all k L 2 (Y X). Proof. The equivalence (i) (ii) was established in the proof of Theorem (iv) (i) Let h T (X, Y ). The functional T k ϕ(x, y)k(y, x)h(x, y)dµ ν(x, y) X Y on C 2 (H 1, H 2 ) is bounded in the operator norm, and has norm not exceeding C. It follows that ϕh T (X, Y ) and ϕh C. Thus, ϕ S(X, Y ) and ϕ S C. (i) (iv) follows from Theorem (ii) (iii) Set K = l 2, a(x) = (a k (x)) k=1 and b(y) = (b k(y)) k=1. (iii) (ii) Let (e k ) k=1 be an orthonormal basis of K and set a k(x) = (a(x), e k ), b k (y) = (e k, b(y)). Then a k (x) 2 = (a(x), e k )(e k, a(x)) = a(x) 2 k=1 k=1 and similarly for b(y); thus the boundedness conditions follow. Similarly, (a(x), b(y)) = (a(x), e k )(e k, b(y)) = a k (x)b k (y) holds for almost all (x, y). k=1 Corollary Every element of S(X, Y ) is equivalent to a (unique) function from C ω (X Y ). Proof. By Theorem 4.13, every element of S(X, Y ) is equivalent, with respect to the product measure on X Y, to a function of the form (x, y) a k (x)b k (y), k=1 k=1

25 HERZ-SCHUR MULTIPLIERS 25 where the sequences (a k ) k=1 L (X, µ) and (b k ) k=1 L (Y, ν) satisfy the conditions in Theorem 4.13 (ii). It is now easy to check that all such functions are ω-continuous. An important subclass of Schur multipliers is formed by the positive ones. A Schur multiplier ϕ S(X, X) is called positive if the map S ϕ is positive, that is, if T B(L 2 (X, µ)), T 0 implies that S ϕ (T ) 0. Exercise Define an order version of the notion of a matricially norming algebra (see Theorem 2.3) and use it to show the following version of R. R. Smith s theorem (Theorem 2.5): If ϕ S(X, X) and S ϕ is positive then S ϕ is completely positive. Exercise Let ϕ S(X, X). The following are equivalent: (i) ϕ is positive; (ii) there exists a separable Hilbert space K and an essentially bounded weakly measurable function a : X K such that ϕ(x, y) = (a(x), a(y)), a.e. on X X. Moreover, if (ii) holds true then ϕ S = esssup x X a(x) Discrete and continuous Schur multipliers. A particular case of special importance is where X and Y are equipped with the counting measure. In this case, it is convenient to drop the assumption on their σ- finiteness, and this consider arbitrary (and not necessarily countable) sets X and Y. Exercise Let X and Y be sets. A function ϕ l (X Y ) is a Schur multiplier with respect to the counting measures on X and Y if and only if (ϕ(x, y)a x,y ) B(l 2 (X), l 2 (Y )) whenever (a x,y ) B(l 2 (X), l 2 (Y )). We include two characterisation results; for their proofs, we refer the reader to [30]. Theorem Let X (resp. Y ) be a locally compact Hausdorff space and µ (resp. ν) be a Radon measure on X (resp. Y ) with support equal to X (resp. Y ). Let ϕ : X Y C be a continuous function. The following are equivalent: (i) ϕ S µ,ν (X, Y ); (ii) ϕ is a Schur multiplier with respect to the counting measures on X and Y. Theorem Let X (resp. Y ) be a locally compact Hausdorff space and µ (resp. ν) be a Radon measure on X (resp. Y ) with support equal to X (resp. Y ). Let ϕ : X Y C be an ω-continuous function. The following are equivalent: (i) ϕ S µ,ν (X, Y ); (ii) there exist null sets M X and N Y such that ϕ (X\M) (Y \N) is a Schur multiplier with respect to the counting measures on X \ M and Y \ N.

26 26 IVAN. TODOROV We finish this section by recalling a well-known example of a function that is not a Schur multiplier. Let X = Y = N, equipped with counting measure. For a number of questions in Operator Theory, it is important to truncate a matrix A = (a i,j ) of an operator in B(l 2 ). In other words, given a subset κ N N, we wish to replace A by the matrix B = (b i,j ), where b i,j = a i,j if (i, j) κ and b i,j = 0 otherwise. If χ κ is a Schur multiplier then B = S χκ (A) and is hence again a bounded operator on l 2. The question which subsets κ have the property that S χκ is a Schur multiplier is still open. (Note that the Schur multipliers that are characteristic functions are precisely the idempotent ones.) The next theorem is often phrased by saying that triangular truncation is unbounded. Theorem Let κ = {(i, j) N N : i j}. Then χ κ is not a Schur multiplier. The theorem has a natural measurable version: Theorem Equip the unit interval [0, 1] with Lebesgue measure and let κ = {(x, y) [0, 1] [0, 1] : x y}. Then χ κ is not a Schur multiplier. While the statement of Theorem 4.20 requires estimates of matrix norms, its measurable version, Theorem 4.21, can be obtained directly using the results of this section; we suggest its proof as an(other) exercise. 5. Further properties of M cb A() 5.1. Embedding into the Schur multipliers. In this section, we establish a fundamental result due to M. Bożejko-. Fendler and J. E. ilbert which establishes an embedding of M cb A() into the algebra of Schur multipliers. Let be a second countable locally compact group equipped with left Haar measure m. We write for short S() = S m,m (, ). Recall the usual notation for the map of conjugation by a unitary operator: if U is a unitary operator acting on a Hilbert space H, we let Ad U (T ) = UT U, T B(H). Let ρ : B(L 2 ()), r ρ r, be the right regular representation of on L 2 (), that is, the representation given by (ρ r f)(s) = (r) 1/2 f(sr), s, r, f L 2 (). We recall that (15) VN() = {ρ s : s }. iven a function h : C and r, let h r : C be given by h r (s, t) = h(sr, tr), s, t. Definition 5.1. A Schur multiplier ϕ S() will be called invariant if S ϕ Ad ρr = Ad ρr S ϕ for every r. We denote by S inv () the set of all invariant Schur multipliers. Lemma 5.2. If ϕ, ψ S() and S ϕ (T ) = S ψ (T ) for all T VN() then ϕ = ψ.