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- W2025867663 abstract "I. Let G be a locally compact group with left invariant Haar measure m. For any measurable subset S of G, define Ls to be that subset of L'(G) consisting of all functions which vanish (a.e.) on the complement of S. When Ls forms an algebra, we call it a vanishing algebra. The notion and study of vanishing algebras was initiated by A. Simon. We recall here some known results (see [2]): (1) If S is a semigroup l.a.e. (i.e., there exists a semigroup T in G such that S= T locally almost everywhere), then Ls is a vanishing algebra. (2) If S is a group l.a.e., then Ls is a self-adjoint vanishing algebra. In this note we shall prove that the converse of (1) is true when G is unimodular and o-compact, and the converse of (2) is always true. We shall first establish some lemmas which are essential in obtaining our results. The main theorems will appear in ??III and IV." @default.
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- W2025867663 date "1963-01-01" @default.
- W2025867663 modified "2023-09-26" @default.
- W2025867663 title "On vanishing algebras" @default.
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