## Approximately multiplicative maps from weighted semilattice algebras

#### Authors

Yemon Choi

Keywords: AMNM, approximate homomorphism, Feinstein algebra, semilattice, weighted convolution algebra.

MSC 2010: primary 39B72; secondary 46J10

DOI: 10.1017/S1446788713000189

#### Status

Copyright is held by the Australian Mathematical Publishing Association Incorporated. They have agreed that I can put a PDF of the final published version on my webpages, so here it is (albeit as the advance copy, without the final page numbers and issue)

Preprint copy of the final version* available at arXiv 1203.6691

(*) That is, the final version modulo journal typesetting, house style, and disagreements about the name of the university overlooking Haymarket metro station in Newcastle, England.

#### Abstract

We investigate which weighted convolution algebras $\ell^1_\omega(S)$, where $S$ is a semilattice, are AMNM in the sense of Johnson (JLMS, 1986). We give an explicit example where this is not the case. We show that the unweighted examples are all AMNM, as are all $\ell^1_\omega(S)$ where $S$ has either finite width or finite height. Some of these finite-width examples are isomorphic to function algebras studied by Feinstein (IJMMS, 1999).

We also investigate when $(\ell^1_\omega(S),{\bf M}_2)$ is an AMNM pair in the sense of Johnson (JLMS, 1988), where ${\bf M}_2$ denotes the algebra of $2$-by-$2$ complex matrices. In particular, we obtain the following two contrasting results: (i) for many non-trivial weights on the totally ordered semilattice ${\bf N}_{\min}$, the pair $(\ell^1_\omega({\bf N}_{\min}),{\bf M}_2)$ is not AMNM; (ii) for any semilattice $S$, the pair $(\ell^1(S),{\bf M}_2)$ is AMNM. The latter result requires a detailed analysis of approximately commuting, approximately idempotent $2\times 2$ matrices.