# Group von Neumann-algebras

Johannes Christensen
Foredrag for studerende
Fredag, 14 februar, 2014, at 15:15-16:00, in Aud. D4 (1531-219)
Abstrakt:
Given a group $G$, one can construct a von Neumann algebra by considering certain bounded operators on the Hilbert space $l^{2}(G)$. These von Neumann algebras are called group von Neumann algebras, and if one works with the restriction of only considering ICC groups, these von Neumann algebras all become factors of type $II_{1}$.

It has been known for a long time, that not all von Neumann algebras constructed using ICC groups are isomorphic, but to which extend the von Neumann algebra inherits the properties of the underlying group is still a rather open question. One example of this is the case where the group is the free group $F_{n}$ on $n>1$ generators. Mathematicians have tried to determine whether these are isomorphic or not for decades, and trying to answer this question, a new branch of mathematics has been constructed - free probability.

In my lecture I will very briefly define a von Neumann algebra and state its basic properties before I begin constructing the group von Neumann algebra, so it should be possible for an audience with no knowlegde of operator-algebras to attend the lecture.
Kontaktperson: Thomas Lundsgaard Schmidt