EXTENDING MULTIPLIERS OF THE FOURIER ALGEBRA FROM A SUBGROUP

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1 EXTENDIN MULTIPLIERS OF TE FOURIER ALEBRA FROM A SUBROUP MICAEL BRANNAN AND BRIAN FORREST Abstract. In this paper, we consider various extension problems associated with elements in the the closure with respect to either the multiplier norm or the completely bounded multiplier norm of the Fourier algebra of a locally compact group. In particular, we show that it is not always possible to extend an element in the closure with respect to the multiplier norm of the Fourier algebra of the free group on two generators, to a multiplier of the Fourier algebra of SL(2, R). 1. Introduction Let be a locally compact group. We let A() and B() denote the Fourier and Fourier-Stieltjes algebras of. These Banach algebras of continuous functions on, introduced in by Eymard in [6], are central objects in non-commutative harmonic analysis. A multiplier of A() is a (necessarily bounded and continuous) function v : C such that va() A(). For each multiplier v of A(), the linear operator M v on A() defined by M v (u) vu for each u A() is bounded via the Closed raph Theorem. The multiplier algebra of A() is the closed subalgebra MA() : {M v : v is a multiplier of A()} of B(A()), where B(A()) denotes the algebra of all bounded linear operators from A() to A(). Throughout this paper we will generally use v in place of the operator M v and we will write v MA() to represent the norm of M v in B(A()). Let be a locally compact group and let V N() its group von Neumann algebra. That is, V N() is the von Neumann algebra generated by the left regular representation λ : U(L 2 ()). The duality A() V N() Key words and phrases. Fourier algebra, multipliers, completely bounded multipliers, locally compact group Mathematics Subject Classification. Primary 43A30, 43A22; Secondary 46L07, 22D25, 22D10. 1

2 2 MICAEL BRANNAN AND BRIAN FORREST equips A() with a natural operator space structure. With this operator space structure we can define the cb-multiplier algebra of A() to be M cb A() : CB(A()) MA(), where CB(A()) denotes the (quantized) Banach algebra of all completely bounded linear maps from A() into itself. We let v Mcb A() denote the cb-norm of the operator M v. It is well known that M cb A() is a closed subalgebra of CB(A()) and is thus a (quantized) Banach algebra with respect to the norm Mcb A(). It is known that in general, and that for v B() Moreover, if v A(), then A() B() M cb A() MA() v B() v Mcb A() v MA(). v A() v B(). In case is an amenable group, we have B() M cb A() MA() and that v B() v Mcb A() v MA() for any v B(). In [7], the first author introduced the following algebra which was denoted in that paper by A M0 (): Definition 1.1. iven a locally compact group, let A cb () def A() M cb A() M cb A(). Clearly, if is amenable, then A() A cb () and the norms agree. owever, in general if is non-amenable, then A() A cb (). Moreover, it can be shown that for a large class of locally compact groups, which contains many interesting non-amenable groups including the free group on two generators F 2 and SL(2, R), that A cb () behaves in much the same manner as does the Fourier algebra of an amenable group [4, 8]. For this class, the so called weakly amenable groups, A cb () has proven to be a rather interesting object to study. In this paper, we will also be interested in multipliers of A() that may or may not be completely bounded but can none the less be approximated by elements of A(). This leads us to the following algebras: Definition 1.2. iven a locally compact group, let A M () def A() MA() MA().

3 EXTENDIN MULTIPLIERS 3 As we have seen, if is amenable, then A() A cb () A M () with equality holding for the various norms. Moreover, in [14] Losert has shown that is amenable if and only if A() A M (). In fact, Losert showed that is amenable whenever the B() and the MA() norms are equivalent on A(). Typically, we have A() A cb () A M (). Since it is well known that M cb (A()) M(A()) for many non-amenable groups, it would be reasonable to speculate that for non-amenable groups that the second inclusion is proper. While this was recently shown in [2] to be the case for the group F 2, in what can only be described as a remarkable result, Losert has shown that for the group SL(2, R), we have M cb A(SL(2, R)) MA(SL(2, R)) and hence that A(SL(2, R)) A cb (SL(2, R)) A M (SL(2, R)). 2. Some general results on restrictions and extensions of completely bounded multipliers Let be a locally compact group and let A() be any of the algebras A(), B(), MA(), M cb A(), A M (), or A cb (). Denote by L g, the left translation operator on A(), by g (i.e. (L g ϕ)(x) ϕ(g 1 x)). Let be a closed subgroup. We will denote by A(), the space of all restrictions of elements of A() to. In each case, it is known that A() A(), and that the restriction map R : A() A() given by is contractive. A natural question arises: R(u) u Question 2.1. For which pairs is the map R : A() A() surjective? Equivalently, for which pairs will it be that every element in A() extends to an element in A(). It is well known, that in general B() B() (see for example [5, Pg. 92]). Since this can happen even when is amenable, we do not expect the restriction mapping to be surjective in general for either of MA() or M cb A(). In stark contrast, erz [11] has shown that for any closed subgroup of any locally compact group A() A(). erz s result (which we refer to as erz s restriction theorem) has proved to be a powerful tool in the study of the Fourier algebra. It s usefulness leads us to ask specifically:

4 4 MICAEL BRANNAN AND BRIAN FORREST Question 2.2. If A() is either A M (), or A cb (), does A() A()? ere it is natural to focus our attention on the case where is nonamenable, since if is amenable, A() A() and erz s restriction theorem establishes the result in the affirmative. In the general case, since A() is always dense in A(), if we could show that the restriction map R has closed range, then we would be done. Now, while we have noted that B() B() in many situations, we at least know that the range of the restriction map is closed. This statement, may well be part of the folklore. A similar result was proved by handehari [9, Lemma 3.2.6] for the algebra B 0 () B() C 0 (). We include the short proof for completeness. Proposition 2.3. For any closed subgroup, B() is a closed subalgebra of B(). Proof. Let π : U( π ) be any weakly continuous unitary representation of. Then we denote by A π (), the B() -closed linear span of the coefficient functions {g π(g)ξ η : ξ, η π } [1]. Let ω : U( ω ) denote the universal unitary representation of. Then B() {g ω(g)ξ η : ξ, η ω }. On the other hand, we have B() {h ω (h)ξ η : ξ, η ω } A ω (), where ω : U( ω ) is the restricted representation. Since A ω () is by B() definition norm closed, then B() A ω (). On the other hand we have A ω () B(). To see this, let v A ω (), then there exist {ξ i } i1, {η i } i1 ω such that i1 ξ i 2 <, i1 ξ i 2 <, and v ω ( )ξ i η i. Clearly satisfies u v. u i1 ω( )ξ i η i A ω () B(), i1 Remark 2.4. The proof above can be clearly modified to show that for any weakly continuous unitary representation π of, we have A π () A π (). In particular, if λ is the left regular repreesntation of, then erz s theorem can be interpreted as A() A λ () A λ () A(). Proposition 2.3 leads naturally to the analogous question for multipliers. Question 2.5. Let be closed. Is the restriction algebra M cb A() always closed in M cb A()? Is MA() always closed in MA()?

5 EXTENDIN MULTIPLIERS 5 Before going on we wish to remark that in the case that is an open subgroup of, then it is a relatively straight forward exercise to show that the answer to the previous question is yes for both the multipliers and the completely bounded mutipliers respectively. Indeed if u M cb A(), then it is easy to show that { u(x) if x, v(x) : 0 if x. is in M cb A(). A similar statement can be made if u MA(). It follows that if is open, then in each case the restriction map is a surjection. 3. [SIN] roups Let be a locally compact group and a closed subgroup. We say that [SIN] if there exists an open neighbourhood base {V } V V of the identity in e which is invariant under inner automorphisms by. In this section, we show that for [SIN], we can provide partial answers to the above questions for c.b. multipliers. In particular, we prove a version of erz s restriction theorem for the algebra A cb (). Our main tool is the following construction due to aagerup and Kraus [10]. Theorem 3.1. ([10, Lemma 1.16]) Let be a closed subgroup of a locally compact group and suppose that the aar modular functions satisfy the relation. Fix ξ C c (). Define a linear map Φ ξ : C b () C b () by the equation Φ ξ (ϕ) ξ (ϕdh) ξ, (ϕ C b ()), where ξ(g) ξ(g 1 ). Then Φ ξ (M cb A()) M cb A(), and ( 2d(g), Φ ξ Mcb A() M cb A() ξ(gh)dh) / where d(g) is the left translation-invariant measure on / induced by aar measure dg in, and normalized so that f(g)dg f(gh)dhd(g), (f C c ()). / Our next result, the main result of this section, is inspired by the work of Cowling and Rodway [5] on extending elements of B() to B() when is a SIN-group and is closed. Theorem 3.2. Let [SIN], and suppose ϕ M cb A() has the property that the map M cb A() h L h ϕ is continuous. Then ϕ M cb A(). Moreover, for any ɛ > 0, there exists u M cb A() such that u ϕ and u Mcb A() (1 + ɛ) ϕ Mcb A().

6 6 MICAEL BRANNAN AND BRIAN FORREST Remark 3.3. Observe that if is discrete and [SIN], then every ϕ M cb A() trivially satisfies the hypothesis of Theorem 3.2, so M cb A() M cb A() in this case. When is not discrete or amenable, it is unknown whether or not every element of M cb A() satisfies the translation-continuity condition of this theorem. Other examples of ϕ M cb A() which do satisfy the hypothesis of Theorem 3.2 are given by coefficient functions of uniformly bounded (not necessarily unitary) representations of, and elements which belong to the closure B() M cb A() M cb A(). Proof. (Of Theorem 3.2.) Fix ϕ M cb A() satisfying the above hypothesis. As in the proof of the open mapping theorem (see for example [17, Theorem 5.9]), it suffices to prove that for any ɛ > 0 there exists u M cb A() with and u Mcb A() ϕ Mcb A(), u ϕ Mcb A() < ɛ. Let V be an open neighbourhood of the identity e such that L h 1ϕ ϕ Mcb A() < ɛ, (h V 1 V ). Let 0 ξ C c () be a function which is -central (i.e. ξ(hxh 1 ) ξ(x) for all h, x ) and suppose suppξ V. This is always possible, since [SIN]. Finally normalize ξ so that ( 2d(g) ξ(gh)dh) 1. / Let u M cb A() be defined by u Φ ξ (ϕ), where Φ ξ is the map given in Theorem 3.1. Then ( 2d(g) u Mcb A() ϕ Mcb A() ξ(gh)dh) ϕ Mcb A(). / We now consider the restriction u M cb A(). For k we have u(k) (ξ (ϕdh) ξ)(k) ξ(g)ϕ(h)ξ(k 1 gh)dhdg ξ(g)ϕ(h)ξ(ghk 1 )dhdg (since ξ is -central) ξ(g)ϕ(hk)ξ(gh)dhdg (since is unimodular) ξ(g)ξ(gh)(l h 1ϕ)(k)dhdg.

7 EXTENDIN MULTIPLIERS 7 On the other hand, note that ξ(g)ξ(gh)dhdg ξ(gh )ξ(gh h)dhdh d(g) / / / ( ξ(gh )ξ(gh)dhdh d(g) ξ(gh)dh) 2d(g) 1. Therefore, for all k, we have u(k) ϕ(k) ξ(g)ξ(gh)[(l h 1ϕ)(k) ϕ(k)]dhdg. Since the function (g, h) v(g)v(gh), is non-negative, continuous, and compactly supported, we can interpret the difference u ϕ as the vector valued integral u ϕ ξ(g)ξ(gh)[l h 1ϕ ϕ]dhdg M cb A(). Furthermore, we have the norm estimate u ϕ Mcb A() ξ(g)ξ(gh) L h 1ϕ ϕ Mcb A()dhdg This completes the proof. sup {(g,h) : v(g)v(gh) 0} L h 1ϕ ϕ Mcb A() sup L h 1ϕ ϕ Mcb A() {g V, h V 1 V } < ɛ. As a consequence of Theorem 3.2, we get an analogue of erz s restriction theorem for A cb () when [SIN]. Corollary 3.4. If [SIN], then the restriction map R : A cb () A cb () is a completely contractive surjection. Proof. It is well known that the restriction map R : M cb A() M cb A() is a complete contraction (see [18, Corollary 6.3]). We therefore only need to show that R : A cb () A cb () is surjective. Let ϕ A cb () and ɛ > 0 be arbitrary. We want to show that ϕ R(A cb ()). Using the same open mapping theorem argument that was used at the start of the proof of Theorem 3.2, our problem reduces to finding u A cb () such that u Acb () ϕ Acb (), and u ϕ Acb () < ɛ. Since ϕ A cb () B() M cb A(), ϕ satisfies the hypothesis of Theorem

8 8 MICAEL BRANNAN AND BRIAN FORREST 3.2 (see Remark 3.3). Let u Φ ξ (ϕ) M cb A() be the c.b. multiplier constructed in the proof of Theorem 3.2. Then u Mcb A() ϕ Acb (), and u ϕ Acb () < ɛ. So to complete the proof, we need to show that u A cb (). This, however, is easy: using the density of A() C c () in A cb (), we can find a sequence {ϕ n } n N A() C c () such that lim ϕ ϕ n Mcb A() 0. n Since Φ ξ : M cb A() M cb A() is a continuous map, which evidently maps compactly supported functions to compactly supported functions, we have u Φ ξ (ϕ) lim Φ Mcb ξ(ϕ n ) M cb A() C c () A() A n }{{} cb (). A() C c() It turns out that when [SIN] and is a discrete subgroup, we can improve on Theorem 3.2 and Corollary 3.4 a bit: Theorem 3.5. Suppose that [SIN] and that is a discrete subgroup. Then there exists an isometry Γ : M cb A() M cb A() which maps A cb () into A cb (), and is a right inverse for the restriction map. That is, Γϕ ϕ for all ϕ M cb A(). An immediate consequence of this theorem is the following: Corollary 3.6. Let be as in Theorem 3.5, and let A() (resp. A()) be either the algebra M cb A() or A cb () (resp. M cb A() or A cb ()). Let I () denote the ideal of functions ψ A(), which vanish on. Then I () is complemented in A(). Furthermore, this is also true with replaced by any set E for which I (E), the ideal of those functions in A() vanishing on E, is complemented in A(). Remark 3.7. We suspect that the above results may well be true with complete isometry replacing isometry and completely complemented replacing complemented. owever, we are at this time unable to prove this assertion. It is also worth noting that the previous two results are new even if is amenable. We will now prove Theorem 3.5. Proof. Choose an open symmetric neighborhood V of the identity e with compact closure such that V V 2 {e}. This can always be done because is a discrete subgroup of. Next, let 0 ξ C c () be a function which is -central (i.e. ξ(hxh 1 ) ξ(x) for all h, x ) and suppose suppξ V. This is possible because [SIN]. Let u ξ ξ A() C c () and normalize ξ so that u(e) ξ 2 L 2 () 1. Note that supp(u) V 2. Define Γ : l () C b ()

9 EXTENDIN MULTIPLIERS 9 by the equation (Γϕ)(x) h ϕ(h)u(h 1 x). Then Γ maps finitely supported functions to compactly supported functions, and Γϕ ϕ for all ϕ l (). To verify this last statement, note that for any h, k, u ξ ˇξ has been chosen so that u(h 1 k) δ h,k. Therefore (Γϕ)(k) h ϕ(h)u(h 1 k) ϕ(k), (ϕ l (), k ). Next we observe that Γ Φ ξ : M cb A() M cb A(). For ϕ M cb A(), and x we have Γϕ(x) h ϕ(h)u(h 1 x) ξ(g)ϕ(h)ξ(x 1 hg)dg h ξ(h 1 g)ϕ(h)ξ(x 1 g)dg h ξ(gh 1 )ϕ(h)ξ(x 1 g)dg h ξ(g)ϕ(h)ξ(x 1 gh)dg h ξ(g)ϕ(h) ξ(h 1 g 1 x)dg h ξ(g)[(ϕdh) ξ](g 1 x)dg [ξ (ϕdh) ξ](x) [Φ ξ (ϕ)](x). So Γ Φ ξ : M cb A() M cb A() and by Theorem 3.1, Γ Φ ξ ( 2d(g) ξ(gh)) / h / h,h / h ξ(gh)ξ(gh )d(g) ξ(gh) 2 d(g) (since g ξ(gh) and g ξ(gh ) have disjoint supports for h h.) ξ(g) 2 dg ξ 2 L 2 () 1.

10 10 MICAEL BRANNAN AND BRIAN FORREST The fact that Γ is an isometry now follows from the contractivity of Γ and the fact that restriction to is a complete contraction from M cb A() to M cb A(). Indeed, for ϕ M cb A(), Γϕ Mcb A() ϕ Mcb A() Γϕ Mcb A() Γϕ Mcb A(). 4. The group SL(2, R) Let s consider now the case where is the connected Lie group SL(2, R), and is a copy of the discrete subgroup F 2. For example, we may take to be the discrete subgroup generated by the (algebraically free) pair of matrices g 1 ( ) ( 1 0, g ). Our main tool will be the following result due to Losert: Theorem 4.1. ([15, Theorem]) For SL(2, R), we have M A() M cb A(), and A() is dense in MA() C 0 () with respect to MA(). For any u MA(), λ lim x u(x) exists, u λ MA() C 0 (), and u MA() u λ MA() + λ. In particular, this result says that A M () A cb () M cb A() C 0 (), and M cb A() C1 1 A cb (). Losert s result gives immediately a negative answer to Question 2.2 for A M () and Question 2.5 for MA() with this choice of and. To see why this is the case, we first need the following proposition. Proposition 4.2. For the free group on two generators F 2, we have that A cb (F 2 ) A M (F 2 ). Proof. Since v Mcb A(F 2 ) v MA(F2 ), if A cb (F 2 ) A M (F 2 ), then it follows immediately that the two norms are equivalent on A(F 2 ). This is impossible since, in [3, Theorem 6.3.3] the first author constructs a set E F 2 such that the ideal I(E) {u A(F 2 ) u(g) 0 for all g E} A(F 2 ) has an approximate identity that is bounded in the the MA(F2 ) norm but not in the Mcb A(F 2 ) norm. Theorem 4.3. The restriction map from A M () to A M () is not surjective, and MA() is not closed in MA(). Proof. Consider the first statement concerning restriction from A M () to A M (). By Theorem 4.1, A M () A cb () M cb A(). Since restriction from to induces a complete contraction from M cb A() into M cb A(), we also have A M () A cb (). On the other hand, for F 2, we

11 EXTENDIN MULTIPLIERS 11 know that A M () A cb (), so A M () A M (). This proves the first statement. Consider now the restriction algebra MA(). Take any element ϕ A M ()\M cb A(). Then, by Theorem 4.1, MA() M cb A(), so On the other hand, we claim that ϕ / MA(). ϕ MA() MA(). To see this, let {ϕ n } n N A() be a sequence such that lim n ϕ n ϕ MA() 0. Then, by erz s restriction theorem, Therefore ϕ MA() MA(). {ϕ n } n N A() MA(). On the level of c.b. multipliers, with the choice of and as above, the situation is quite different than it is for [SIN] groups (c.f. Theorem 3.5). Theorem 4.4. The restriction map from M cb A() to M cb A() is not surjective. Proof. Let g 1, g 2 be the free generators of F 2. Then the set E {g1 ng 2g1 n } n N is algebraically free. It follows from the work of Leinert [13] (or more recently [16, Theorem 0.1]), that {ϕ M cb A() : suppϕ E} l (E), completely isomorphically. Now choose any ϕ l (E) such that lim n ϕ(g1 ng 2g1 n ) does not exist. (For example, take ϕ(g1 ng 2g1 n ) ( 1)n ). We claim that there is no u M cb A() such that u ϕ. Indeed, if such a u existed, Theorem 4.1 would imply that lim u(x) λ C. x But since this would mean that which is a contradiction. g n 1 g 2 g n 1, (n ), lim n ϕ(gn 1 g 2 g1 n ) lim n u(gn 1 g 2 g1 n ) λ, Remark 4.5. At this point we do not know if M cb A(SL(2, R)) F2 is closed in M cb A(F 2 ), and hence whether or not A cb (F 2 ) A cb (SL(2, R)) F2? We note that it may well be possible that equality does hold above. One suggestive piece of evidence in this direction arises from the representation theorem for c.b. multipliers. Let ξ L 2 () be any unit vector, and let (ω, ω ) be the universal unitary representation of. Examining the proof

12 12 MICAEL BRANNAN AND BRIAN FORREST of Jolissaint [12], we see that for ϕ M cb A(), there exist bounded maps V 1, V 2 : L 2 () ω such that ϕ Mcb A() V 1 V 2, and such that ϕ is represented by the twisted coefficient function (4.1) ϕ(y 1 x) ω(x)v 1 λ(x 1 )ξ ω(y)v 2 λ(y 1 )ξ, (x, y ). So any restriction, ψ M cb A() will be a twisted coefficient associated to ω. This is somewhat reminiscent of the situation for B(), though of course we are lacking anything as complete as Arsac s work on B() [1] in our understanding of M cb A() to help us complete the argument References [1]. Arsac, Sur l espaces de Banach engendré par les coefficients d une représentation unitaire., Publ. Dép. Math. (Lyon), 13 (1976) [2] M. Brannan, B. Forrest and C. Zwarich, Multipliers, completely bounded multipliers, and ideals in the Fourier algebra. Preprint, [3] M. Brannan, Operator spaces and ideals in Fourier algebras. Thesis, University of Waterloo, [4] M. Cowling and U. aagerup, Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one. Invent. Math. 96 (1989), [5] M. Cowling and P. Rodway, Restrictions of certain function spaces to closed subgroups of locally compact groups. Pacific J. Math. 80 (1979), no. 1, [6] P. Eymard, L algèbre de Fourier d un groupe localement compact. Bull. Soc. Math. France 92 (1964), [7] B. Forrest, Some Banach algebras without discontinuous derivations. Proc. Amer. Math. Soc.114 (1992), [8] B. Forrest, V. Runde and N. Spronk, Operator amenability of the Fourier algebra in the cb-multiplier norm. Canadian J. Math. 59 (2007), [9] M. handehari, armonic analysis of Rajchman algebras. Thesis, University of Waterloo, [10] U. aagerup and J. Kraus Approximation properties for group C algebras and group von Neumann algebras. Trans. Amer. Math. Soc. 344 (1994), no. 2, [11] C. erz, armonic synthesis for subgroups. Annales de l Institut Fourier, tome 23, no. 3, renoble (1973), [12] P. Jolissaint, A characterization of completely bounded multipliers of Fourier algebras. Colloq. Math. 63 (1992), no. 2, [13] M. Leinert, Faltungsoperatoren auf gewissen diskreten ruppen. Studia Math. 52 (1974), [14] V. Losert, Properties of the Fourier algebra that are equivalent to amenability. Proc. Amer. Math. Soc. 91(1984), [15] V. Losert, On multipliers and completely bounded multipliers - the case SL(2, R). Preprint, [16]. Pisier, Multipliers and lacunary sets in non-amenable groups. Amer. J. Math. 117 (1995), no. 2, [17] W. Rudin, Real and complex analysis. Mcraw-ill, Third Ed., [18] N. Spronk, Measurable Schur multipliers and completely bounded multipliers of the Fourier algebras. Proc. London Math. Soc. 89 (1): , 2004.

13 EXTENDIN MULTIPLIERS 13 Michael Brannan: Department of Mathematics and Statistics, Queen s University, 99 University Avenue, Kingston, Ontario, Canada, K7L 3N6. mbrannan@mast.queensu.ca Brian Forrest: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada, N2L beforres@math.uwaterloo.ca