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Weak Convergence Is Not Strong Convergence For Amenable Groups

Published online by Cambridge University Press:  20 November 2018

Joseph M. Rosenblatt  and
George A. Willis
Joseph M. Rosenblatt
Affiliation:
Department of Mathematics University of Illinois at Urbana Urbana, Illinois 61801 U.S.A., e-mail: jrsnbltt@math.uiuc.edu
George A. Willis
Affiliation:
Department of Mathematics University of Newcastle Callaghan, NSW 2308, Australia, e-mail: george@frey.newcastle.edu.au
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Abstract

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Let $G$ be an infinite discrete amenable group or a non-discrete amenable group. It is shown how to construct a net $\left( {{f}_{\alpha }} \right)$ of positive, normalized functions in ${{L}_{1}}\left( G \right)$ such that the net converges weak* to invariance but does not converge strongly to invariance. The solution of certain linear equations determined by colorings of the Cayley graphs of the group are central to this construction.

Keywords

43A07
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 44 , Issue 2 , 01 June 2001 , pp. 231 - 241
Copyright
Copyright © Canadian Mathematical Society 2001

References

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