Vol. 18, No. 2, 1966

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Point-determining homomorphisms on multiplicative semi-groups of continuous functions

Robert Stephen De Zur

Vol. 18 (1966), No. 2, 227–236

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.

Mathematical Subject Classification
Primary: 46.25
Received: 4 October 1964
Published: 1 August 1966
Robert Stephen De Zur