Brown's Spectral Distribution Measure for R-Diagonal Elements in Finite von Neumann Algebras

✍ Scribed by Uffe Haagerup; Flemming Larsen

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 287 KB
• Volume: 176
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.2000.3610

✦ Synopsis

In 1983 L. G. Brown introduced a spectral distribution measure for non-normal elements in a finite von Neumann algebra M with respect to a fixed normal faithful tracial state {. In this paper we compute Brown's spectral distribution measure in case T has a polar decomposition T=UH where U is a Haar unitary and U and H are V-free. (When Ker T=[0] this is equivalent to that (T, T*) is an R-diagonal pair in the sense of Nica and Speicher.) The measure + T is expressed explicitly in terms of the S-transform of the distribution + T *T of the positive operator T *T. In case T is a circular element, i.e., T=(X 1 +iX 2 )Â-2 where (X 1 , X 2 ) is a free semicircular system, then sp T=D , the closed unit disk, and + T has constant density 1Â? on D .