GLEASON-KAHANE-ZELASKO TYPE THEOREMS FOR COMPLEX RIESZ ALGEBRAS

Let U be a complex f-algebra with a unit element e. It is shown that a linear functional f on U is a lattice homomorphism with f(e) = 1 if and only if f(a) is an element of sigma(a) for all a is an element of U. More generally, let U be a complex Riesz algebra with a positive unit element e. It turns out that the principal band B-e in U generated by e is a projection band in U. Moreover, a linear functional f on U is a lattice homomorphism with f(e) = 1 if and only if f(a) is an element of sigma(P-e(a)) for all a is an element of U, where P-e denotes the band projection of U onto B-e. It follows that if E is a Dedekind complete complex Riesz space then a linear functional f on L-r(E) is an identity preserving lattice homomorphism if and only if for each T is an element of L-r(E) the scalar f(T) is a spectral value in L(E) of the diagonal component D(T) of T.