Zero product preserving maps on Banach algebras of Lipschitz functions
Let \((K,d)\) a nonempty compact metric space and let \(\alpha\in ]0,1[\) The authors consider the Banach algebras of complex-valued Hölder functions of order \(\alpha,\) \[ \text{Lip}_\alpha=\{f:K\to \mathbb{C}: p_\alpha(f):=\sup_{x,y\in K, x\neq y} \frac{|f(x)-f(y)|}{d(x,y)^\alpha}<+\infty\} \] and \[ \text{lip}_\alpha=\{f\in\text{Lip}_\alpha\; | \lim_{d(x,y)\to 0} \frac{f(x)-f(y)}{d(x,y)^\alpha}=0 \}, \] endowed with the norm given by \(\|f\|_\infty + p_\alpha(f)\). Using a nice argument related to some Fourier type algebra of continuous functions on the unit circle, the authors prove that any Banach space valued continuous bilinear mapping \(\varphi\) on \(\text{Lip}_\alpha\) (resp. \(\text{lip}_\alpha\)) such that \(\varphi(f,g)=0\) for \(f,g \in\text{Lip}_\alpha\) (resp., \(\text{lip}_\alpha\)) with \(fg=0\) must satisfy the identity \[ \varphi(f,g)+\varphi(g,f)=\varphi(fg,\mathbf{1}) +\varphi(\mathbf{1},fg). \] This result is used to deduce that any separating linear map \(T:\text{Lip}_\alpha\to B\) onto a commutative unital Banach algebra \(B\) is of the form \(T=w\Phi,\) where \(w\) is an invertible element in \(B\) and \(\Phi:\text{Lip}_\alpha\to B\) is an onto homomorphism. Here separating means that \(T(f)T(g)=0\) if \(fg=0\). The analogous result holds for \(\text{lip}_\alpha\).
scientific article; zbMATH DE number 5731667
scientific article