An extension of a non-commutative Choquet-Deny theorem
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Abstract:
Let $G$ be a discrete group, and let $N$ be a normal subgroup of $G$. Then the quotient map $G\to G/N$ induces a group algebra homomorphism $T_N:\ell ^1(G)\to \ell ^1(G/N)$. It is shown that the kernel of this map may be decomposed as $\ker (T_N)=R+L$, where $R$ is a closed right ideal with a bounded left approximate identity and $L$ is a closed left ideal with a bounded right approximate identity. It follows from this fact that, if $I$ is a closed two-sided ideal in $\ell ^1(G)$, then $T_N(I)$ is closed in $\ell ^1(G/N)$. This answers a question of Reiter.References
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Additional Information
- G. A. Willis
- Affiliation: Department of Mathematics, The University of Newcastle, New South Wales, Australia, 2308
- MR Author ID: 183250
- Email: george@frey.newcastle.edu.au
- Received by editor(s): June 24, 1995
- Received by editor(s) in revised form: September 5, 1995, and March 10, 1998
- Published electronically: March 3, 1999
- Communicated by: Palle E. T. Jorgensen
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 111-118
- MSC (1991): Primary 43A20; Secondary 22D40
- DOI: https://doi.org/10.1090/S0002-9939-99-05117-5
- MathSciNet review: 1637448