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A universal semigroup

Published online by Cambridge University Press:  09 April 2009

J. H. Michael
Affiliation:
University of AdelaideSouth Australia
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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].

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1970

References

[1]Kakutani, S., ‘Simultaneous extension of continuous functions considered as a positive liner operation’, Japanese Journal of Mathematics 17 (1940), 14.CrossRefGoogle Scholar
[2]Kelly, J. L., General topology (Van Nostrand, 1955).Google Scholar
[3]Riesz, F. and Sz-Nagy, B., Functional analysis (Ungar, 1955).Google Scholar
[4]Ulam, S. M., A collection of mathematical problems (Interscience, 1960).Google Scholar