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Using very original methods from operator algebras, Sorin Popa has shown that the orbit structure of the Bernoulli action of a property (T) group, completely remembers the group and the action. This information is even essentially contained in the crossed product von Neumann algebra. This is the first von Neumann strong rigidity theorem in the literature. The same methods allow Popa to obtain II $_{1}$ factors with prescribed countable fundamental group.

Semiregular maximal abelian *-subalgebras and the solution to the factor state Stone-Weierstrass problem.

S. Popa (1984)

Inventiones mathematicae

Simplicity of the projective unitary groups defined by simple factors.

P. de de Harpe (1979)

Commentarii mathematici Helvetici

Solution of the factorial Stone-Weierstrass conjecture. An application of the theory of standard split W*-inclusions.

R. Longo (1984)

Inventiones mathematicae

Sommes de commutateurs dans les algèbres de von Neumann finies continues

Thierry Fack, Pierre de La Harpe (1980)

Annales de l'institut Fourier

Soit $M$ une algèbre de von Neumann finie. Nous montrons que l’espace des sommes finies de commutateurs de $M$ coïncide avec le noyau de la trace centrale. Si $M$ est un facteur, il en résulte par exemple que tout élément est une combinaison linéaire finie de projecteurs de dimension $1 / 2$ . Nous montrons aussi dans ce cas que le groupe dérivé de $G L (M)$ coïncide avec le noyau du déterminant de Fuglede-Kadison.

Stability of $C^{*}$ -algebras is not a stable property.

Rørdam, Mikael (1997)

Documenta Mathematica

Stable rank and real rank of compact transformation group C*-algebras

Robert J. Archbold, Eberhard Kaniuth (2006)

Studia Mathematica

Let (G,X) be a transformation group, where X is a locally compact Hausdorff space and G is a compact group. We investigate the stable rank and the real rank of the transformation group C*-algebra C₀(X)⋊ G. Explicit formulae are given in the case where X and G are second countable and X is locally of finite G-orbit type. As a consequence, we calculate the ranks of the group C*-algebra C*(ℝⁿ ⋊ G), where G is a connected closed subgroup of SO(n) acting on ℝⁿ by rotation.

Standard and split inclusions of von Neumann algebras.

S. Doplicher, R. Longo (1984)

Inventiones mathematicae

Subalgebras of a Finite Algebra.

Erik Christensen (1979)

Mathematische Annalen

Subtriviality of continuous fields of nuclear C*-algebras.

Etienne Blanchard (1997)

Journal für die reine und angewandte Mathematik

The action of groups on real factors of type III $_{λ}$ , $λ \neq 1$ .

Rakhimov, A.A. (2004)

Zapiski Nauchnykh Seminarov POMI

The alternative Dunford-Pettis Property in the predual of a von Neumann algebra

Miguel Martín, Antonio M. Peralta (2001)

Studia Mathematica

Let A be a type II von Neumann algebra with predual A⁎. We prove that A⁎ does not have the alternative Dunford-Pettis property introduced by W. Freedman [7], i.e., there is a sequence (φₙ) converging weakly to φ in A⁎ with ||φₙ|| = ||φ|| = 1 for all n ∈ ℕ and a weakly null sequence (xₙ) in A such that φₙ(xₙ) ↛ 0. This answers a question posed in [7].

The C*-algebra generated by two projections.

Allan M. Sinclair, Iain Raeburn (1989)

Mathematica Scandinavica

The Dimension Lattice of a JBW-Algebra.

Klaus Alvermann (1987)

Mathematische Zeitschrift

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