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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.”

##### MSC:
 46J10 Banach algebras of continuous functions, function algebras
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