  In this note we present two new positive answers to Tingley's problem in
certain subspaces of function algebras. In the first result we prove that every
surjective isometry between the unit spheres, $S(A)$ and $S(B)$, of two
uniformly closed function algebras $A$ and $B$ on locally compact Hausdorff
spaces can be extended to a surjective real linear isometry from $A$ onto $B$.
In a second goal we study surjective isometries between the unit spheres of two
abelian JB$^*$-triples represented as spaces of continuous functions of the
form $$C^{\mathbb{T}}_0 (X) := \{ a \in C_0(X) : a (\lambda t) = \lambda a(t)
\hbox{ for every } (\lambda, t) \in \mathbb{T}\times X\},$$ where $X$ is a
(locally compact Hausdorff) principal $\mathbb{T}$-bundle. We establish that
every surjective isometry $\Delta: S(C_0^{\mathbb{T}}(X))\to
S(C_0^{\mathbb{T}}(Y))$ admits an extension to a surjective real linear
isometry between these two abelian JB$^*$-triples.
