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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Local existence of $\mathcal{K}$-sets, projective tensor products, and Arens regularity for $A(E_{1}+\dots +E_{n})$


Author: Colin C. Graham
Translated by:
Journal: Proc. Amer. Math. Soc. 132 (2004), 1963-1971
MSC (2000): Primary 43A15, 43A10; Secondary 46L10
Published electronically: February 6, 2004
MathSciNet review: 2053967
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Abstract: Theorem. If $X_{1},\dots ,X_{n}$ are perfect compact subsets of the locally compact metrizable abelian group, then there are pairwise disjoint perfect subsets $Y_{1}\subseteq X_{1},\dots ,Y_{n}\subseteq X_{n}$such that (i) $Y_{j}$ is either a Kronecker set or (ii) for some $p_{j}\ge 2$, $Y_{j}$ is a translate of a $K_{p_{j}}$-set all of whose elements have order $p_{j}$, and (iii) $A(Y_{1}+\dots +Y_{n})$ is isomorphic to the projective tensor product $C(Y_{1}) \hat \otimes \cdots \hat \otimes C(Y_{n})$.

This extends what was previously known for groups such as $\mathbb{T}$ or for the case $n=2$to the general locally compact abelian group. Old results concerning the local existence of Kronecker and $K_{p}$-sets are improved.


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Additional Information

Colin C. Graham
Affiliation: Department of Mathematics, University of British Columbia, RR#1–H-46, Bowen Island, British Columbia, Canada V0N 1G0
Email: ccgraham@alum.mit.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07159-X
PII: S 0002-9939(04)07159-X
Keywords: Arens regularity, bidual, Kronecker set, $K_{p}$-set, locally compact abelian groups, projective tensor product, quotients of the Fourier algebra, set sums, tensor algebra
Received by editor(s): September 12, 2002
Received by editor(s) in revised form: December 23, 2002
Published electronically: February 6, 2004
Additional Notes: Preprints of a draft of this paper were circulated under the title “Arens regularity and related matters for $A(E+F)$”.
Communicated by: Andreas Seeger
Article copyright: © Copyright 2004 American Mathematical Society