Abstract
Li et al. [‘Weak 2-local isometries on uniform algebras and Lipschitz algebras’, Publ. Mat.63 (2019), 241–264] generalized the Kowalski–Słodkowski theorem by establishing the following spherical variant: let A be a unital complex Banach algebra and let
$\Delta : A \to \mathbb {C}$
be a mapping satisfying the following properties:
(a)
$\Delta $
is 1-homogeneous (that is,
$\Delta (\lambda x)=\lambda \Delta (x)$
for all
$x \in A$
,
$\lambda \in \mathbb C$
);
(b)
$\Delta (x)-\Delta (y) \in \mathbb {T}\sigma (x-y), \quad x,y \in A$
.
Then
$\Delta $
is linear and there exists
$\lambda _{0} \in \mathbb {T}$
such that
$\lambda _{0}\Delta $
is multiplicative. In this note we prove that if (a) is relaxed to
$\Delta (0)=0$
, then
$\Delta $
is complex-linear or conjugate-linear and
$\overline {\Delta (\mathbf {1})}\Delta $
is multiplicative. We extend the Kowalski–Słodkowski theorem as a conclusion. As a corollary, we prove that every 2-local map in the set of all surjective isometries (without assuming linearity) on a certain function space is in fact a surjective isometry. This gives an affirmative answer to a problem on 2-local isometries posed by Molnár [‘On 2-local *-automorphisms and 2-local isometries of B(H)', J. Math. Anal. Appl.479(1) (2019), 569–580] and also in a private communication between Molnár and O. Hatori, 2018.
On the open ball centered at an invertible element of a Banach algebra
