On spectra and Brown's spectral measures of elements in free products of matrix algebras

• Autores: Junsheng Fang, Don Hadwin, Xiujuan Ma
• Localización: Mathematica scandinavica, ISSN 0025-5521, Vol. 103, Nº 1, 2008, págs. 77-96
• Idioma: inglés
• DOI: 10.7146/math.scand.a-15070
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• Resumen

  • We compute spectra and Brown measures of some non self-adjoint operators in (M2(C), 1 2 Tr) .

    (M2(C), 1 2 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), 1 2 Tr) . 1 and 1 . (M2(C), 1 2 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 = ¥ë1, then X has a nontrivial hyperinvariant subspace affiliated with (M2(C), 1 2 Tr) . (M2(C), 1 2 Tr).