Let X and Y be compact
Hausdorff spaces, C(X) and C(Y ) the algebras of real valued continuous functions on
X and Y respectively with the usual sup norms. If T is an algebra homomorphism
from C(X) onto a dense subset of C(Y ) then by a theorem of Stone, T induces a
homeomorphism μ from Y to X and it necessarily follows that Tf(y) = 0 if and only
if f(μ(y)) = 0.
In a more general setting, viewing C(X) and C(Y ) as multiplicative semi-groups,
let T be a semi-group homomorphism from C(X) onto a dense point-separating set in
C(Y ). No such map μ satisfying the above condition need exist. T is called
point-determining in case for each y there is an x such that Tf(y) = 0 if and only if
f(x) = 0. It is shown that such a homomorphism T induces a homeomorphism from
Y into X in such a way that Tf(y) = [sgnf(x)]|f(x)|p(x) for some continuous
positive function p where x is related to y via the induced homeomorphism, that such
a T is an algebra homomorphism followed by a semi-group automorphism, and that
T is continuous.
