The Atiyah conjecture and Artinian rings

Details The Atiyah conjecture and Artinian rings The Atiyah conjecture and Artinian rings

2007-11-21

arxiv.org/abs/0711.3328v1

Pure and Applied Mathematics Quarterly 8 (2012), no. 2, 313--328

Mathematics

Rings and Algebras

10 pages, submitted to the proceedings of the Morelia conference in honor of Farrell-Jones

Scientific paper

Let G be a group such that its finite subgroups have bounded order, let d denote the lowest common multiple of the orders of the finite subgroups of G, and let K be a subfield of C that is closed under complex conjugation. Let U(G) denote the algebra of unbounded operators affiliated to the group von Neumann algebra N(G), and let D(KG,U(G)) denote the division closure of KG in U(G); thus D(KG,U(G)) is the smallest subring of U(G) containing KG that is closed under taking inverses. Suppose n is a positive integer, and \alpha \in \Mat_n(KG). Then \alpha induces a bounded linear map \alpha: l^2(G)^n \to \l^2(G)^n, and \ker\alpha has a well-defined von Neumann dimension \dim_{N(G)} (\ker\alpha). This is a nonnegative real number, and one version of the Atiyah conjecture states that d \dim_{N(G)}(\ker\alpha) \in Z. Assuming this conjecture, we shall prove that if G has no nontrivial finite normal subgroup, then D(KG,U(G)) is a d \times d matrix ring over a skew field. We shall also consider the case when G has a nontrivial finite normal subgroup, and other subrings of U(G) that contain KG.

Affiliated with

Also associated with

No associations

Rating

The Atiyah conjecture and Artinian rings does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with The Atiyah conjecture and Artinian rings, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The Atiyah conjecture and Artinian rings will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-414310