Shizuo Kakutani

Ring and lattice characterizations of complex Hilbert space

SHIZUO KAKUTANI AND GEORGE W. MACKEY Introduction. In an earlier paper [l] by the authors it was suggested that at least the ring characterization of real Hubert space given therein might be extended to the complex case by making use of a device employed by B. H. Arnold [2] in so extending a theorem of Eidelheit. It is the purpose of the present note to show that this can indeed be done not only for the ring characterization but for the lattice one as well. The difficulty in the complex case is that the complex field admits a great many discontinuous automorphisms. It is overcome by making use of the device of Arnold mentioned above to show that in the infinite-dimensional case only continuous automorphisms present themselves (see Lemma 2 below). It is shown by an example that the infinite-dimensionality is essential and that accordingly the theorems of [l] cannot be extended to the complex case in quite their full generality.

SOME MULTIPLICATIVE LINEAR FUNCTIONALS ON M(G)

acter on point measures. In the present paper, we replace the independent sets P by Cantor sets S containing many dependent elements. The behavior of multiplicative linear functionals for measures with support contained in the sets S is far more restricted than is the case for independent Cantor sets. Nevertheless some very curious multiplicative linear functionals can be constructed. Details are given in ? 4; the general theorem on which our constructions depend appears in ? 3. The present paper was suggested by the interesting paper [4] of Yu. A.