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

Authors

  • Junsheng Fang
  • Don Hadwin
  • Xiujuan Ma

DOI:

https://doi.org/10.7146/math.scand.a-15070

Abstract

We compute spectra and Brown measures of some non self-adjoint operators in (M2(C),12Tr)(M2(C),12Tr), 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),12Tr)1 and 1(M2(C),12Tr), 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 Xλ1, then X has a nontrivial hyperinvariant subspace affiliated with (M2(C),12Tr)(M2(C),12Tr).

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Published

2008-09-01

How to Cite

Fang, J., Hadwin, D., & Ma, X. (2008). On spectra and Brown’s spectral measures of elements in free products of matrix algebras. MATHEMATICA SCANDINAVICA, 103(1), 77–96. https://doi.org/10.7146/math.scand.a-15070

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Articles