Approximately multiplicative maps from weighted semilattice algebras

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.