A universal semigroup

Extract

In [4] S. Ulam asks the following question.‘Does there exist a universal compact semigroup; i.e., a semigroup U such that every compact topological semigroup is continuously isomorphic to a subsemigroup of it?’The author has not been able to answer this question. However, in this paper, a proof is given for the following related result.

Let Q denote the Hilbert cube of countably infinite dimension and C(Q) the Banach space of continuous real-valued functions on Q with the usual norm. Let U denote the semigroup consisting of all bounded linear operators T: C(Q)→ C(Q) with ∥T∥ ≦ 1 and let U be endowed with the strong topology. Then, for every compact metric semigroup S with the property:(1.1) for all x, y ∈ S, with x ≠ y, there exists a z ∈ S, such that xz ≠ yz or zx ≠ zy;

there exists a 1 − 1 mapping φ of S into U such that φ is both a semigroup isomorphism and a homeomorphism.

U is metrizable, but is not compact; hence it does not provide an answer to the question of Ulam. The proof of the above statement leans heavily on a result of S. Kakutani [1].