Celebratio Mathematica — Singer

I. M. Singer: a brief biography

Works connected to Richard V. Kadison

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R. V. Kadison and I. M. Singer: “Extensions of pure states,” Amer. J. Math. 81 : 2 (April 1959), pp. 383–400. MR 123922 Zbl 0086.09704 article

The main concern of this paper is the problem of uniqueness of extensions of pure states from maximal abelian self-adjoint algebras of operators on a Hilbert space to the algebra of bounded operators on that space. The answer, as many of have suspected for several years, is in the negative. To the best of our knowledge, the problem is not recorded in the literature. We heard of it first from I. E. Segal and I. Kaplansky, though it is difficult to credit a problem which stems naturally from the physical interpretation and the inherent structure of a subject. This problem has arisen, in one form or another, in our work on several different occasions; and we have been gathering bits of information related to it, over the years. (The present solution is prompted by just such a reappearance.)