Let \(A\) be a complex Banach algebra. For \(a,b\in A\), define the two-sided multiplication \(M_{a,b}\) on \(A\) by \(M_{a,b}: x\mapsto axb\). Recall that \(A\) is said to be prime if \(M_{a,b}=0\) implies that \(a=0\) or \(b=0\) for any pair of elements \(a\) and \(b\) in \(A\). Define \(\kappa(A)=\inf\{\|M_{a,b}\|:a,b\in A,\;\|a\|=\|b\|=1\}\) and call \(A\) ultraprime if \(\kappa(A)>0\). This notion was introduced in the reviewer’s PhD thesis [Tübingen (1986)] and published in [M. Mathieu, Proc. Cent. Math. Anal. Aust. Natl. Univ. 21, 297–317 (1989; Zbl 0701.46027)]. It has successfully served as a concept that allows to transfer algebraic techniques into an analytic context since \(A\) is ultraprime if and only if every ultrapower of \(A\) (with respect to a free ultrafilter on \(\mathbb N\)) is prime.
In the paper under review, the authors use this methodology to investigate approximately commuting linear mappings as well as approximate Lie isomorphisms and derivations.
Let \(T: A\to A\) be a linear mapping and set \(\text{com}(T)=\sup\{\|T(x)x-xT(x)\|:x\in A,\;\|x\|=1\}\). Then \(T\) is called commuting if \(\text{com}(T)=0\). The closed linear subspace of the Banach algebra \(\mathcal L(A)\) of all bounded linear mappings on \(A\) that consists of the commuting mappings will be denoted by \(\text{LCom}(A)\). Evidently, for each \(T\in\mathcal L(A)\), \(\text{com}(T)\leq2\,\text{dist}\bigl(T,\text{LCom}(A)\bigr)\), the distance from \(T\) to \(\text{LCom}(A)\). Using various facts on ultraproducts and that every ultraprime Banach algebra is centrally closed [loc.cit.], the authors obtain in their first main result the following.
Theorem 2.5. For each \(K>0\), there exists \(M>0\) such that \(\text{dist}\bigl(T,\text{LCom}(A)\bigr)\leq M\,\text{com}(T)\) for every Banach algebra \(A\) with \(\kappa(A)\geq K\) and \(T\in\mathcal L(A)\).
A like result is obtained for commuting quadratic mappings in Theorem 2.8.
In Sections 3 and 4 of the paper, these results are then used to describe small perturbations of Lie isomorphisms and Lie derivations on ultraprime Banach algebras with emphasis on the case \(A=\mathcal L(H)\), \(H\) a Hilbert space.