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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.”

46J10 Banach algebras of continuous functions, function algebras
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