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- W2038448026 abstract "Introduction. Let G be a compact abelian group with character group r. For basic definitions the reader is advised to see Chapters 1 and 9 of Rudin [7]. Each closed subalgebra A of the group algebra L1(G) induces an equivalence relation on F: fig if and only if f(ac)=f(i) everyf in A, wheref is the Fourier transform of f. We will denote the equivalence classes of by {EA} where A is in some index set and distinguish one special class EO = {y E : f(y) = 0 for every f in A} called the zero set of A. Since r is discrete and Je C0(F) for every f in L1(G), each EA, A#=O, is finite. EO may be infinite. For A =0 let PA be the trigonometric polynomial whose transform is the characteristic function of EA, and Ao the smallest closed algebra containing every PA. Rudin [7, p. 231] has shown that Ao induces and is contained in every closed subalgebra which induces with zero set EO. If we define AO to be the algebra of all functions whose transforms are constant on every E1A and zero on EO then AO will be the largest closed algebra inducing . Rudin [7] asked if there exist distinct closed subalgebras which induce the same equivalence relation. Or, equivalently, does there exist a closed subalgebra A with AO = AO? Kahane [4] gives a negative answer for G = T, the circle group. In ?1 we give sufficient conditions for an equivalence relation on the integers Z to be uniquely induced by exactly one closed subalgebra of L1(T). This result strengthens Theorem 3 of Kahane [4]. We also prove his result on Z x Z. In ?2 we study the algebras Ao and AO in detail and give necessary and sufficient conditions for AO=AA. Finally in ?3 we will consider factorization in closed subalgebras of L1(G) and show that although L1(G) has factorization, there exists a closed subalgebra A without factorization and such that A n LP(G) is L1-dense in A, where 1 <p < 2." @default.
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- W2038448026 date "1970-01-01" @default.
- W2038448026 modified "2023-09-26" @default.
- W2038448026 title "Closed subalgebras of group algebras" @default.
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- W2038448026 doi "https://doi.org/10.1090/s0002-9947-1970-0254522-2" @default.
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