Approximately multiplicative maps from weighted semilattice algebras

Abstract

We investigate which weighted convolution algebras ${\ell }_{\omega }^1(S)$, 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 ${\ell }_{\omega }^1(S)$ 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 $({\ell }_{\omega }^1(S),{\mathbf{M}}_2)$ is an AMNM pair in the sense of Johnson (JLMS, 1988), where ${\mathbf{M}}_2$ 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 ${\mathbf{N}}_{\min }$, the pair $({\ell }_{\omega }^1({\mathbf{N}}_{\min }),{\mathbf{M}}_2)$ is not AMNM; (ii) for any semilattice $S$, the pair $({\ell }^1(S),{\mathbf{M}}_2)$ is AMNM. The latter result requires a detailed analysis of approximately commuting, approximately idempotent $2\times 2$ matrices.

AMS-LaTeX. v3: 31 pages, additional minor corrections to v2. Final version, to appear in J. Austral. Math. Soc. v4: small correction of mis-statement at start of Section 4 (this should also be fixed in the journal version)

Cited by in corpus (4)

• Stability of characters and filters for weighted semilattices
• Unavoidable subprojections in union-closed set systems of infinite breadth
• Constructing non-AMNM weighted convolution algebras for every semilattice of infinite breadth
• Approximately multiplicative maps between algebras of bounded operators on Banach spaces