ON SPECTRA AND BROWN'S SPECTRAL MEASURES OF ELEMENTS IN FREE PRODUCTS OF MATRIX ALGEBRAS

We compute spectra and Brown measures of some non self-adjoint operators in (M2(C), ½ Tr) * (M2(C), ½ Tr), the reduced free product von Neumann algebra of M2(C) with M2(C). Examples include AB and A + B, where A and B are matrices in (M2(C), ½ Tr) * 1 and 1 * (M2(C), ½ Tr), respectively. We prove th...

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Bibliographic Details
Published in:Mathematica scandinavica 2008-01, Vol.103 (1), p.77-96
Main Authors: FANG, JUNSHENG, HADWIN, DON, MA, XIUJUAN
Format: Article
Language:English
Subjects:
Algebra
Eigenvalues
Ellipses
Mathematical theorems
Random variables
Scalars
Semimajor axis
Semiminor axis
Von Neumann algebra
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Summary:We compute spectra and Brown measures of some non self-adjoint operators in (M2(C), ½ Tr) * (M2(C), ½ Tr), the reduced free product von Neumann algebra of M2(C) with M2(C). Examples include AB and A + B, where A and B are matrices in (M2(C), ½ Tr) * 1 and 1 * (M2(C), ½ Tr), respectively. We prove that AB is an R-diagonal operator (in the sense of Nica and Speicher [12]) if and only if Tr(A) = Tr(B) = 0. We show that if X = AB or X = A + B and A, B are not scalar matrices, then the Brown measure of X is not concentrated on a single point. By a theorem of Haagerup and Schultz [9], we obtain that if X = AB or X = A + B and X ≠ λ1, then X has a nontrivial hyperinvariant subspace affiliated with (M2(C), ½ Tr) * (M2(C), ½ Tr).
ISSN:0025-5521
1903-1807
DOI:10.7146/math.scand.a-15070