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Transactions of the American Mathematical Society

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Projections and approximate identities for ideals in group algebras


Authors: Teng-Sun Liu, Arnoud van Rooij and Ju Kwei Wang
Journal: Trans. Amer. Math. Soc. 175 (1973), 469-482
MSC: Primary 43A20
DOI: https://doi.org/10.1090/S0002-9947-1973-0318781-2
MathSciNet review: 0318781
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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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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1973-0318781-2
Keywords: Continuous projections, bounded approximate identities, closed left ideals, closed right ideals, closed 2-sided ideals, group algebras, modular functions, property $ {{\text{P}}_1}$, $ {w^ \ast }$-closed, Calderón set, spectral set, discrete coset ring, factorable closed ideals, Bohr compactification
Article copyright: © Copyright 1973 American Mathematical Society

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