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An extension of a non-commutative
Choquet-Deny theorem

Author: G. A. Willis
Journal: Proc. Amer. Math. Soc. 128 (2000), 111-118
MSC (1991): Primary 43A20; Secondary 22D40
Published electronically: March 3, 1999
MathSciNet review: 1637448
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Abstract | References | Similar Articles | Additional Information

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.

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Additional Information

G. A. Willis
Affiliation: Department of Mathematics, The University of Newcastle, New South Wales, Australia, 2308

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
Article copyright: © Copyright 1999 American Mathematical Society