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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Group extensions and cohomology for locally compact groups. IV

Author: Calvin C. Moore
Journal: Trans. Amer. Math. Soc. 221 (1976), 35-58
MSC: Primary 22D05; Secondary 22D10, 22D30
MathSciNet review: 0414776
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Abstract: In this paper we shall apply the cohomology groups constructed in [14] to a variety of problems in analysis. We show that cohomology classes admit direct integral decompositions, and we obtain as a special case a new proof of the existence of direct integral decompositions of unitary representations. This also leads to a Frobenius reciprocity theorem for induced modules, and we obtain splitting theorems for direct integrals of tori analogous to known results for direct sums. We also obtain implementation theorems for groups of automorphisms of von Neumann algebras. We show that the splitting group topology on the two-dimensional cohomology groups agrees with other naturally defined topologies and we find conditions under which this topology is $ {T_2}$. Finally we resolve several questions left open concerning splitting groups in a previous paper [13].

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Article copyright: © Copyright 1976 American Mathematical Society