Li et al. [‘Weak 2local 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 1homogeneous (that is,
$\Delta (\lambda x)=\lambda \Delta (x)$
for all
$x \in A$
,
$\lambda \in \mathbb C$
);

(b)
$\Delta (x)\Delta (y) \in \mathbb {T}\sigma (xy), \quad x,y \in A$
.
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 complexlinear or conjugatelinear and
$\overline {\Delta (\mathbf {1})}\Delta $
is multiplicative. We extend the Kowalski–Słodkowski theorem as a conclusion. As a corollary, we prove that every 2local 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 2local isometries posed by Molnár [‘On 2local *automorphisms and 2local 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.