Celebratio Mathematica

R. V. Kadison and I. M. Singer: “Some remarks on representations of connected groups,” Proc. Nat. Acad. Sci. U. S. A. 38 : 5 (May 1952), pp. 419–423. MR 48456 Zbl 0046.25202 article

The purpose of this note is to bring to light a fact which has escaped notice, viz., in the direct integral reduction of the regular representation of a connected separable locally compact group, factors of Type \( \mathrm{II}_1 \) occur almost nowhere. This proof is carried out by the following scheme of argument. We show first that a connected locally compact group which has sufficiently many unitary representations which generate rings of finite type is the group direct product of a compact group and an abelian group. From this it follows quite easily that a unitary representation of a connected locally compact group generates a ring of operators which has no summand of Type \( \mathrm{II}_1 \) and, in particular, is not itself a factor of Type \( \mathrm{II}_1 \). Employing a theorem of Mautner, to the effect that, for almost every factor in the direct integral reduction of the regular representation of a group, there exists a strongly continuous representation of the group which generates the factor, we obtain the final result.

R. V. Kadison and I. M. Singer: “Three test problems in operator theory,” Pac. J. Math. 7 : 2 (February 1957), pp. 1101–1106. MR 92123 Zbl 0078.11503 article

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.)

R. V. Kadison and I. M. Singer: “Triangular operator algebras: Fundamentals and hyperreducible theory,” Amer. J. Math. 82 : 2 (April 1960), pp. 227–259. MR 121675 Zbl 0096.31703 article

R. V. Kadison: “Which Singer is that?,” pp. 347–373 in Papers dedicated to Atiyah, Bott, Hirzebruch and Singer (Cambridge, MA, 14–16 May 1999). Surveys in Differential Geometry 7. International Press (Somerville, MA), 2000. Reprinted in The founders of index theory (2009). MR 1919431 Zbl 1075.01010 incollection

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R. V. Kadison: “Which Singer is that?,” pp. 279–302 in The founders of index theory: Reminiscences of Atiyah, Bott, Hirzebruch, and Singer. Edited by S.-T. Yau. International Press (Somerville, MA), 2009. Originally published in Surveys in Differential Geometry 7 (2000). incollection

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Raoul H. Bott	Related	Volume
Shing-Tung Yau	Related

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Isadore M. Singer

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