Projections and approximate identities for ideals in group algebras

Liu, Teng-Sun; van Rooij, Arnoud; Wang, Ju Kwei

doi:10.1090/S0002-9947-1973-0318781-2

Projections and approximate identities for ideals in group algebras
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by Teng-Sun Liu, Arnoud van Rooij and Ju Kwei Wang PDF

Trans. Amer. Math. Soc. 175 (1973), 469-482 Request permission

Abstract:

For a locally compact group G with property $({{\text {P}}_1})$, if there is a continuous projection of ${L^1}(G)$ onto a closed left ideal I, then there is a bounded right approximate identity in I. If I is further 2-sided, then I has a 2-sided approximate identity. The converse is proved for ${w^ \ast }$-closed left ideals. Let G be further abelian and let I be a closed ideal in ${L^1}(G)$. The condition that I has a bounded approximate identity is characterized in a number of ways which include (1) the factorability of I, (2) that the hull of I is in the discrete coset ring of the dual group, and (3) that I is the kernel of a closed element in the discrete coset ring of the dual group.

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