By Antoine Derighetti (auth.)

This quantity is dedicated to a scientific research of the Banach algebra of the convolution operators of a in the community compact crew. encouraged through classical Fourier research we ponder operators on Lp areas, arriving at an outline of those operators and Lp models of the theorems of Wiener and Kaplansky-Helson.

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See [36], J. Dixmier, Chap. I, Sect. 3, no. 4, Corollaire 1, p. 42. The next result is Kaplansky’s density theorem. Theorem 2. H/ with B C. H/. S˛ / of B such that: 1: lim˛ S˛ D T strongly, 2: kS˛ k Ä kT k for every ˛. Proof. See Dixmier, [36], Chap. I, Sect. 3, no. 5, Th´eor`eme 3, p. 43–44. Let G be a locally compact group. In this paragraph, we denote by A the set of all /, where is a complex measure with finite support. G// with unit: 2G . / D 2G . G/ . The C following statement is straightforward.

Iv. F / for ˛ 2 C and F 2 A0p , p j. / D G . F /jjjp D kF kA0p for F 2 A0p . In particular the map j is injective. We verify only (iii). According to Proposition 2 we have Á D lL D k but E p 0 G . A0p ; Ap /. Suppose that E; F are two complex vector spaces and ˇ B a bilinear form on n ˇ 0 E F . L; u/ D 0 for every o L 2 M and similarly for a subset of F . M 0 /0 . Suppose now that M is a subspace of E. G/, ˇ with M . G . G/, and therefore u D 0. We have proved that M 0 D 0 and consequently that M 00 D E.

G/. s / Ä jjjS jjjp Np0 . x/dx. f /. Proposition 1. G/, s 2 S Œg and r 2 T Œf . Then: D T p G. s . G/ and Proof. G/. g/Œ' and D I D T p G . pf Then by Theorem 1 of Sect. x/. x/dx. G/ Then by Theorem 7 of Sect. G/ and . g/Œ'; Œ . x/dx: Q G Proposition 2. G/, s 2 S Œg and r 2 T Œf . g/T G . G/ and every Proof. G/. p G . g/T p G. pf E /Œ'; Œ : 0 Then by Theorem 7 of Sect. s G D I D T and therefore p G. pf /Œ'; Œs E : / We finally apply Theorem 1 of Sect. s . x/dx: G In the following it will be decisive to assume the unimodularity of the locally compact group G.