@phdthesis{Deprez2015a,
author = {Deprez, Tobe},
school = {KU Leuven},
title = {Relations between inner amenability, property {$\Gamma$}, stability and McDuff},
type = {Master thesis},
year = {2015},
abstract = {This master thesis discusses links between two mathematical structures: groups and von Neumann algebras. Groups originated from the study of symmetries of polygons, polyhedra and other geometrical objects. Consider, for example, the symmetries of a square: one can rotate over 90, 180 and 270 degrees, reflect across the two diagonals and across the lines through the midpoints of opposite edges, or leave it unchanged (mathematicians call this last 'symmetry' the identity transformation). Composing two symmetries (i.e. applying one after the other), yields another symmetry. For example, first rotating the square over 90 degrees, and then reflecting it across a diagonal, yields the same transformation as a reflection across a line through the midpoints of two opposite edges. In general, the order of the operation matters: a reflection across a diagonal, followed by a rotation over 90 degrees, gives another symmetry than first applying the rotation and then the reflection. However, there are groups where the order of composition doesn't matter. The symmetry group of the flag of the United Kingdom is such an example. This group consists of four elements: the identity transformation, the reflections across the horizontal and vertical axis, and a rotation over 180 degrees. A group with this property is called commutative. A von Neumann algebra is another structure which was introduced by Murray and von Neumann in the '30s and '40s of the 20th century. The main motivation for their introduction was quantum mechanics, but von Neumann algebras are now also used in other areas of mathematics and physics. Starting from a group, one can construct two von Neumann algebras: the 'group von Neumann algebra' and 'the crossed product von Neumann algebra' (actually the latter is a von Neumann algebra constructed from an 'action' of a group, rather than from the group itself). A central question is how the properties of a group are related to the properties of its von Neumann algebras. One can for example prove that the group von Neumann algebra of a commutative group is again commutative, while the crossed product von Neumann algebra is not. It is known that if two groups are very similar, then their group von Neumann algebras are similar, while on the other hand, two group von Neumann algebras can be very similar, even though their underlying groups are quite different. In my master thesis, I study the relations between four properties: two properties on the level of groups (inner amenability and stability), one property of the group von Neumann algebra (property Gamma) and one of the crossed product (the McDuff property). The relations between these properties are very subtle. For example, it is known since 1975 that if a group von Neumann algebra has property Gamma, then the underlying group must be inner amenable. But, the converse was only solved very recently in the negative by Vaes, i.e. he was able to prove that there are inner amenable groups such that their group von Neumann algebras do not have property Gamma. It turns out that inner amenability is the weakest of the four properties, i.e. stable groups are inner amenable, and if a group von Neumann algebra has property Gamma or a crossed product is McDuff, then the underlying group must be inner amenable. Also, crossed products of stable groups are McDuff. But, none of the other implications hold.}
}