×

zbMATH — the first resource for mathematics

Approximately multiplicative maps from weighted semilattice algebras. (English) Zbl 1317.46033
The author investigates the AMNM problem, first studied by B. E. Johnson [J. Lond. Math. Soc., II. Ser. 34, 489–510 (1986; Zbl 0625.46059)]: When is an approximately multiplicative map near multiplicative? Roughly speaking, a linear map \(\psi\) is approximately multiplicative if \(\psi(xy)-\psi(x)\psi(y)\) is small, and it is near multiplicative if \(\psi\) is close in norm to a multiplicative mapping (i.e., a homomorphism). In this paper, the author concentrates on the case where the domain is a (weighted) convolution algebra on a semilattice. From the author’s overview:
“We then observe that \(\ell^1(S)\) is AMNM for any semilattice \(S\) (Theorem 3.1). On the other hand, we give an explicit example of a semilattice \(T\) and a weight on \(T\) such that the weighted convolution algebra \(\ell^1_\omega(T)\) is not AMNM (Theorem 3.4). In Section 3.3, as a special case of a general technical result, we prove that if \(S\) has either finite width or finite height, then \(\ell^1_\omega(T)\) is AMNM for every weight \(\omega\). This applies in particular when \(S = {\mathbb N}_{\min}\) the original case of interest.
The picture is far less complete if we consider approximately multiplicative maps into algebras other than \(\mathbb C\). Let \({\mathbb T}_2\) be the (commutative, non-semisimple) algebra of dual numbers over \(\mathbb C\), and let \({\mathbb M}_2\) be the (non-commutative, semisimple) algebra of \(2\times 2\) matrices with entries in \(\mathbb C\). In Theorem 4.2, we show that whenever \(\omega\) is a non-trivial weight on \({\mathbb N}_{\min}\), then \((\ell^1_\omega({\mathbb N}_{\min}),{\mathbb T}_2)\) is not an AMNM pair. In Theorem 4.7, we show that for many non-trivial weights on \({\mathbb N}_{\min}\), the pair \((\ell^1_\omega({\mathbb N}_{\min}), {\mathbb M}_2)\) fails to be AMNM.
These examples suggest that, if we want positive AMNM results for range algebras other than \(\mathbb C\), we should focus attention on the unweighted case. Indeed, we prove that for an arbitrary semilattice \(S\), the pairs \((\ell^1(S),{\mathbb T}_2)\) and \((\ell^1(S), {\mathbb M}_2)\) are ‘uniformly AMNM’ (the terminology is explained below, in Definition 2.4). The proof of this for \({\mathbb M}_2\) takes up all of Section 5: although the techniques used are elementary, a complete proof seems to require substantially more work than is needed for \({\mathbb T}_2\). Finally, we close the paper by briefly discussing some possible avenues for future work.”

MSC:
46J10 Banach algebras of continuous functions, function algebras
PDF BibTeX Cite
Full Text: DOI arXiv
References:
[1] Hewitt, Trans. Amer. Math. Soc. 83 pp 70– (1956)
[2] DOI: 10.1016/j.jfa.2007.12.011 · Zbl 1146.46023
[3] DOI: 10.1155/S0161171299224374 · Zbl 0978.46033
[4] Dales, Dissertationes Math. (Rozprawy Mat.) 474 pp 58– (2010)
[5] Birkhoff, Lattice Theory (1979)
[6] DOI: 10.1112/S0024609396002585 · Zbl 0865.46035
[7] DOI: 10.1007/BF02189397 · Zbl 0594.46047
[8] DOI: 10.1112/jlms/s2-37.2.294 · Zbl 0652.46031
[9] DOI: 10.1112/jlms/s2-34.3.489 · Zbl 0625.46059
[10] Jarosz, Studia Math. 124 pp 37– (1997)
[11] Jarosz, Perturbations of Banach Algebras (1985) · Zbl 0557.46029
[12] DOI: 10.1112/S0024610703004551 · Zbl 1064.46032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.