Abstract
The author recently “proved”, using a result of Wong [4, Corollary I. (3)], that the dimension two normal cohomology of a semifinite von Neumann algebra is trivial. However, it was apparent that one could by the same means prove that every von Neumann algebra possessed “property P” [2, Definition I] despite Schwartz’s example [2, 71, based on the free group with two generators, of a von Neumann algebra that does not have property P. Indeed, one could show that any discrete group is amenable. Evidently Wong’s theorem is wrong. The error lies in appealing to Hall’s marriage theorem [4, p. 160; 3, p. 5231 without verifying that the hypotheses of that theorem are satisfied. A counterexample will be constructed using the following result, which is an abstraction of the proof of [2, Lemma 51; see also [I, Proposition 4.4.151. The corollary is an immediate application to amenability of groups.
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