Uniqueness of the injective \(III_ 1\) factor (Q1187718)

The present Lecture Notes volume collects together the mathematics which went into U. Haagerup's recent and remarkable proof that the injective (hyperfinite) \(III_ 1\) factor is unique. The author states in the introduction that his ``exposition contains nothing that is not already in (the papers) [10] and [16], but merely fills in details in some of the arguments appearing there''. These papers are: \textit{A. Connes}, J. Oper. Theory 14, 189-211 (1985; Zbl 0597.46063) and \textit{U. Haagerup}, Acta Math. 158, 95-148 (1989; Zbl 0628.46061). The uniqueness question dates back to von Neumann's original papers, and it had remained open up to the remarkable paper by U. Haagerup. Connes had shown that its solution depends on a certain bicentralizer problem. The latter was solved, and the uniqueness established, in Haagerup's paper. This is a remarkable break-through and certainly justifies a set of lecture notes of the present form. Recall that Murray and von Neumann had shown in the Thirties that the study of rings of operators (i.e., von Neumann algebras) may be reduced, to some extend, to the more elementary rings, the factors. The classification types I, II and III were defined, and further divided into \(II_ 1\), \(II_{\infty}\), \(III_{\lambda}\), \(0\leq \lambda \leq 1\). The hyperfinite factors were further isolated, and von Neumann showed uniqueness of the hyperfinite \(II_ 1\) (i.e., precisely one isomorphism class). Physicists discovered that type III is dictated by relativistic quantum field theory, and the uniqueness question for the hyperfinite \(III_ 1\) assumed more relevance and significance. Another important milestone was Connes' proof of the equivalence of hyperfiniteness and injectivity in the Seventies, and an elegant self- contained proof by U. Haagerup of the same equivalence in the Eighties.