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    Rights statement: http://journals.cambridge.org/action/displayJournal?jid=AJZ The final, definitive version of this article has been published in the Journal, Journal of the Australian Mathematical Society, 95 (1), pp 36-67 2013, © 2013 Cambridge University Press.

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Approximately multiplicative maps from weighted semilattice algebras

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Published
<mark>Journal publication date</mark>08/2013
<mark>Journal</mark>Journal of the Australian Mathematical Society
Issue number1
Volume95
Number of pages32
Pages (from-to)36-67
Publication StatusPublished
<mark>Original language</mark>English

Abstract

We investigate which weighted convolution algebras ℓ1ω(S), where S is a semilattice, are AMNM in the sense of Johnson [‘Approximately multiplicative functionals’, J. Lond. Math. Soc. (2) 34(3) (1986), 489–510]. We give an explicit example where this is not the case. We show that the unweighted examples are all AMNM, as are all ℓ1ω(S) where S has either finite width or finite height. Some of these finite-width examples are isomorphic to function algebras studied by Feinstein [‘Strong Ditkin algebras without bounded relative units’, Int. J. Math. Math. Sci. 22(2) (1999), 437–443]. We also investigate when (ℓ1ω(S),M2) is an AMNM pair in the sense of Johnson [‘Approximately multiplicative maps between Banach algebras’, J. Lond. Math. Soc. (2) 37(2) (1988), 294–316], where M2 denotes the algebra of 2×2 complex matrices. In particular, we obtain the following two contrasting results: (i) for many nontrivial weights on the totally ordered semilattice Nmin, the pair (ℓ1ω(Nmin),M2) is not AMNM; (ii) for any semilattice S, the pair (ℓ1(S),M2) is AMNM. The latter result requires a detailed analysis of approximately commuting, approximately idempotent 2×2 matrices.

Bibliographic note

http://journals.cambridge.org/action/displayJournal?jid=AJZ The final, definitive version of this article has been published in the Journal, Journal of the Australian Mathematical Society, 95 (1), pp 36-67 2013, © 2013 Cambridge University Press.