D'Alembert's functional equation on topological monoids

We prove that if f is a continuous complex-valued function on the topological monoid M with neutral element e satisfying the functional equation f (xyz) + f (xzy) = 2f(x)f(yz) + 2f(y)f(zx) + 2f(z)f(xy) - 4f(x)f(y)f(z) and f(e) = 1, then there is a continuous homomorphism h : M -> Mat(2)(C), the multiplicative monoid of complex 2 x 2 matrices such that f = 1/2tr o h. As a consequence we prove that if f is a continuous function on the topological group G satisfying f(xy) + f(xy(-1)) = 2f(x)f(y) and f(e) = 1 then there is a continuous homomorphism h : G -> SL(2)(C) such that f = 1/2tr o h,