Chapter 2 The Commutation s Theorem

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1 Chapter 2 The Commutation s Theorem We show that for a locally compact unimodular group, everyt 2 CV 2./ is the limit of convolution operators associated to bounded measures. 2.1 The Convolution Operator T p.f / Theorem 1. Let be a locally compact group, 1 < p < 1, T 2 CV p./, f 2 M 1 00./, r 2 TŒf, ' 2 Lp./ and 2 L p0./. Then: 1: Qr 2 L p0./, 2: N p 0. Qr/ jjjt jj p N p 0. / jf.x/j.x/ 1 p 0 dx, 3: T p. pf /Œ' ; Œ '.x/. Qr/.x/dx. Proof. To begin with suppose ' 2 C 00./. Wehave T p. pf /Œ' ; Œ Œ' r ;Œ : From.j'jjrj/j j2l 1./ we get.' r/.x/.x/dx '.x/. Qr/.x/dx: The inequalities '.x/. Qr/.x/dx ˇ ˇ N p.' r/n p 0. / jjjt jj p N p.'/n p 0. / f.x/j.x/ 1 p 0 dx prove.1/ and.2/. A. erighetti, Convolution Operators on roups, Lecture Notes of the Unione Matematica Italiana 11, OI / , Springer-Verlag Berlin Heidelberg

2 26 2 The Commutation s Theorem Suppose now that ' 2 L p./. There is a sequence.' n / of C 00./ with N p.' n '/! 0.Wehave lim ' n.x/. Qr/.x/dx T p. pf /Œ' ; Œ and ˇ '.x/. Qr/.x/dx ' n.x/. Qr/.x/dx ˇ N p.' n '/N p 0. Qr/: Consequently '.x/. Qr/.x/dx T p. pf /Œ' ; Œ : Remark. ven for p 2, we are unable to decide whether j jjlrj is in L p0./. We now show that every T 2 CV p./ can be approximated by T p.f /. Proposition 2. Let be a locally compact group, 1<p<1and I the set of all f 2 C 00./ with f.x/ 0 for every x 2, f.e/ 6 0 and f.x/.x/ 1=p0 dx 1: Then: 1: on I the relation supp f 0 supp f is a filtering partial order, 2: for f 2 I we have ˇˇˇˇˇˇp. 1, pf/ˇˇˇˇˇˇp 3: for every T 2 CV p./ the net T p. pf/ converges strongly to T. Proof. Let T 2 CV p./, ' 2 L p./ and ">0.LetU be a neighborhood of e in such that for y 2 U f 2I N p '.'/ y 1.y 1 / 1=p < Let also f 2 I with supp f U.From ktœ' T p. pf/œ' k p jjjt jj p we get ktœ' T p. pf/œ' k p <". ".1 C jjjt jj p / : N p '.'/ y 1.y 1 / 1=p f.y/.y 1 / 1=p0 dy

3 2.1 The Convolution Operator T p.f / 27 The investigation of CV 2./ requires the study of those continuous operators S of L 2./ for which S.' a /.S'/ a. In full analogy with Sect. 1.2 we have. '/.x/ '.y 1 x/d.y/ for 2 M 1./, ' 2 C 00./ and x 2.Wealsohave ' 2 C./\ L p./ and N p. '/ kkn p.'/ for 1<p<1. There is a unique continuous operator S of L p./ with SŒ' Œ ' for ' 2 C 00./. WehaveS.f a /.Sf / a for f 2 L p./ and a 2. This operator S is denoted p./. Forf 2 L1./ we set p.f / p.f m / and p.œf / p.f /. efinition 1. Let be a locally compact group, 1<p<1and S 2 L.L p.//. We say that S belongs to the set CV d p./ if S.' a/.s'/ a for every a 2 and for every ' 2 L p./. Proposition 3. Let be a locally compact group and 1<p<1. ThenCV d p./ is a Banach subalgebra of L.L p.//. Proposition 4. Let be a locally compact group and 1<p<1. Then: 1: p is a linear injective contraction of the Banach space M 1./ into the Banach space CV d p./, 2: for every a 2 and every ' 2 L p./ we have p.ı a/' a 1' and ˇˇˇˇˇˇp.ı 1, a/ˇˇˇˇˇˇp 3: p.ı ab/ p.ı a/ p.ı b/ for every a; b 2, 4: for f 2 L 1./ and ' 2 C 00./ we have p.f /Œ' f Œ'. Theorem 5. Let be a locally compact group 1<p<1and S 2 L.L p.//. Then S 2 CV d p./ if and only if S ' 1=p0 f.s'/ 1=p0 f for every f 2 L 1./ and every ' 2 L p./. Remarks. 1: The map x 7! 2.ı x/ is the left regular representation of. 2: The proofs of Proposition 4 and Theorem 5 are entirely similar to those of the corresponding results concerning CV p./ and p.cf Sect. 1.2/.

4 28 2 The Commutation s Theorem Similarly to Theorem 1 and Proposition 2 the following two results are verified. Proposition 6. Let be a locally compact group, 1<p<1and I the set of all f 2 C 00./ with f.x/ 0 for every x 2, f.e/ 6 0 and f.x/dx 1. Then: 1: on I the relation supp f 0 supp f is a filtering partial order, 2: For f 2 I we have ˇˇˇˇˇˇp.f /ˇˇˇˇˇˇp 1, 3: For every S 2 CV d p./ the net S p f.f / converges strongly to S. 2I Theorem 7. Let be a locally compact group, 1 < p < 1, S 2 CV d p./, g 2 M 1 00./, s 2 SŒg, ' 2 Lp./ and 2 L p0./. Then: 1: s 2 L p0./, 2: N p 0.s / jjjsjj p N p 0. /N 1.g/, 3: S p.g/œ' ; Œ '.x/.s /.x/dx. 2.2 A Commutation Property of CV 2./ For S 2 CV d p./ and T 2 CV p./, we first obtain integral formulas for T p.f /Sp.g/ and for Sp.g/T p.f /. Proposition 1. Let be a locally compact group, 1<p<1, S 2 CV d p./, T 2 CV p./, f; g 2 M 1 00./, s 2 SŒg and r 2 TŒf. Then: T p. pf/s p.g/œ' ; Œ '.x/.s. Qr//.x/dx for ' 2 L p./ and 2 L p0./. Proof. Let ' 1 2 S p.g/œ' and I T p. pf/s p.g/œ' ; Œ : Then by Theorem 1 of Sect. 2.1 Qr 2 L p0./ and I Consequently I S p.g/œ' ; Œ Qr : ' 1.x/. Qr/.x/dx.

5 2.2 A Commutation Property of CV 2./ 29 Then by Theorem 7 of Sect. 2.1 s. Qr/ 2 L p0./ and S p.g/œ' ; Œ Qr '.x/.s. Qr//.x/dx: Proposition 2. Let be a locally compact group, 1<p<1, S 2 CV d p./, T 2 CV p./, f; g 2 M 1 00./, s 2 SŒg and r 2 TŒf. Then: S p.g/t p. pf /Œ' ; Œ '.x/..s / Qr/.x/dx for every ' 2 L p./ and every 2 L p0./. Proof. Let ' 1 2 T p. pf/œ' and I S p.g/t p. pf /Œ' ; Œ : Then by Theorem 7 of Sect. 2.1 s 2 L p0./ and I ' 1.x/.s /.x/dx and therefore I T p. pf /Œ' ; Œs : We finally apply Theorem 1 of Sect. 2.1: we have.s / Qr 2 L p0./ and I '.x/.s. Qr//.x/dx: In the following it will be decisive to assume the unimodularity of the locally compact group. With this assumption, we have p f f L. Lemma 3. Let be a locally compact unimodular group, S 2 CV d 2./, T 2 CV 2./ and f; g 2 M 1 00./.ThenT2.f /S2.g/ S2.g/T 2.f /. Proof. For r 2 TŒf, s 2 SŒg and '; 2 M 1 00./ we have T 2. f/s L 2.g/Œ' ; Œ '.x/.s. Qr//.x/dx and S 2.g/T 2. f L /Œ' ; Œ '.x/..s / Qr/.x/dx:

6 30 2 The Commutation s Theorem By the unimodularity of, foreveryx 2 we have.j jjlrj/ x 2 L 2./, and consequently 0 1 j.yxz/jr.z/jd za dy < 1: This implies s. Qr/.s / Qr. Theorem 4. Let be a locally compact unimodular group. Then ST TS for S 2 CV d 2./ and T 2 CV 2./. Proof. To begin with we prove that for S 2 CV d 2./, T 2 CV 2./ and f 2 M 1 00./ we have ST2.f / T2.f /S. Let ' 2 L 2./ and ">0.Thereisg 2 C 00./ with: g.x/ 0 for every x 2, g.x/dx 1, ks 2.g/' S'k " 2 < 2.1 C ˇˇˇˇˇˇT 2.f /ˇˇˇˇˇˇ2/ Now from and ks 2.g/T 2.f /' ST2.f /'k 2 < " 2 : kst 2.f /' T2.f /S'k 2 kst 2.f /' S2.g/T 2.f /'k 2 CkS 2.g/T 2.f /' T2.f /S2.g/'k 2 CkT 2.f /S2.g/' T2.f /S'k 2; Lemma 3 and kt 2.f /S2.g/' T2.f /S'k 2 < " 2 we get kst 2.f /' T2.f /S'k 2 <": Next let ' 2 L 2./ and ">0. According to Proposition 2 of Sect. 2.1 there is f 2 C 00./ with f.x/ 0 for every x 2, f.x/dx 1, kts' T 2.f /S'k 2 < " 2 and kt' T 2.f /'k 2 < " 2.1 C jjjsjj 2 / : From kts' ST'k 2 kts' T 2.f /S'k 2CkT 2.f /S' ST2.f /'k 2CkST 2.f /' ST'k 2; T 2.f /S' ST2.f /' and kst2.f /' ST'k 2 < " 2 we obtain kts' ST'k 2 <":

7 2.3 An Approximation Theorem for CV 2./ An Approximation Theorem for CV 2./ Using the commutation theorem of Sect. 2.2 (Theorem 4) we show that every T 2 CV 2./ is the limit of 2./ for a locally compact unimodular group. For a complex Hilbert space H, we denote by L.H/ the involutive Banach algebra of all continuous operators of H. ForT 2 L.H/, kt k is the norm of the operator T.For a subset of L.H/ we denote by 0 the set of all T 2 L.H/ with ST TS for every S 2, and we put / 0. Theorem 1. n Let H be a complex o Hilbert space and B an involutive subalgebra of ˇ L.H/ with Txˇx 2 H; T 2 B dense in H.ThenB 00 coincides with the closure of B in L.H/ with respect to the strong operator topology. Proof. See [36], J. ixmier, Chap. I, Sect. 3, no. 4, Corollaire 1, p. 42. The next result is Kaplansky s density theorem. Theorem 2. Let H beacomplexhilbertspaceandb; C two involutivesubalgebras of L.H/ with B C. Suppose that C is dense in the strong closure of B in L.H/. Then for every T 2 C there is a net.s / of B such that: 1: lim S T strongly, 2: ks k kt k for every. Proof. See ixmier, [36], Chap. I, Sect. 3, no. 5, Théorème 3, p Let be a locally compact group. In this paragraph, we denote by A the set of all 2./,whereisacomplex measure with finite support. Clearly A is an involutive subalgebra L.L 2.// with unit: 2./ 2. Q/ and 2.ı e/ id L 2 C./.The following statement is straightforward. Proposition 3. Let be a locally compact group. Then CV d 2./ A0. We obtain now the promised approximation theorem for CV 2./. Theorem 4. Let be a locally compact unimodular group and T 2 CV 2./. There is a net. / of complex measures with finite support such that: 1: lim ˇ 2. / T strongly, 2: ˇˇˇˇˇ2 jjjt jj. /ˇˇˇˇˇˇ2 2 for every. Proof. By Theorem 4 of Sect. 2.2 we have T 2 A 00. It suffices to apply Theorems 1 and 2 to finish the proof. n Remarks. 1: The fact that 2.ı ˇ 00 x/ ˇx 2 o CV 2./, for locally compact and unimodular, is due to Segal.[110], Theorem, p. 294/. The case of discrete, was obtained earlier by Murray and von Neumann n. [96], Lemma 5.3.3, p. 789/. 2: Using different methods, ixmier obtained 2.ı ˇ 00 x/ ˇx 2 o CV2./, and consequently Theorem 4, for every locally compact group.[35], Théorème 1,

8 32 2 The Commutation s Theorem p. 280, [36], Chap. I, Sect. 5, p. 71, Théorème 1 and xercice 5 p. 80/. See also Mackey.[90], p. 207, Lemma 3.3./ Theorem 5. Let be a locally compact unimodular group and T 2 CV 2./. There is a net.f / of C 00./ such that: 1: lim ˇ 2.f / T strongly, 2: ˇˇˇˇˇ2 jjjt jj.f /ˇˇˇˇˇˇ2 2 for every. 00 Proof. According to Theorem //.C isthe strong closure of 2.C 00.//. 0 But by Theorem 5 of Sect //.C CV d 2./ and consequently 2.C 00.// 00 CV2./: Remark. We will extend this result to p 6 2 for certain classes of locally compact groups. We will also try to give more information on the approximating net.f /.

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