A Random Matrix Approach to the Lack of Projections in C*_red(F_2)

By Uffe Haagerup, Hanne Schultz, S. Thorbjørnsen. 2004

52 citations recorded by the source corpus.

In 1982 Pimsner and Voiculescu computed the K_0- and K_1-groups of the reduced group C*-algebra C*_red(F_k) of the free group F_k on k generators and settled thereby a long standing conjecture: C*_red(F_k) has no projections except for the trivial projections 0 and 1. Later simpler proofs of this conjecture were found by methods from K-theory or from non-commutative differential geometry. In this paper we provide a new proof of the fact that C*_red(F_k) is projectionless. The new proof is based on random matrices and is obtained by a refinement of the methods recently used by the first and the third named author to show that the semigroup Ext(C*_red(F_k)) is not a group for k >= 2. By the same type of methods we also obtain that two phenomena proved by Bai and Silverstein for certain classes of random matrices: ``no eigenvalues outside (a small neighbourhood of) the support of the limiting distribution'' and ``exact separation of eigenvalues by gaps in the limiting distribution'' also hold for arbitrary non-commutative selfadjoint polynomials of independent GUE, GOE or GSE random matrices with matrix coefficients.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a975b11bbe0a21589de36758e78c9056a3a02822

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In Geometry and Topology

  1. Ramon van Handel
  2. Roland Speicher
  3. Afonso S. Bandeira
  4. Moritz Weber
  5. Sven Raum

In Mathematical Physics

  1. Afonso S. Bandeira
  2. March Boedihardjo
  3. Ramon van Handel
  4. Roland Speicher
  5. Moritz Weber
  6. Sven Raum

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