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

Junsheng Fang·Don Hadwin·Xiujuan Ma
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Abstract

We compute spectra and Brown measures of some non self-adjoint operators in $(M_2(\cc), {1/2}Tr)*(M_2(\cc), {1/2}Tr)$, the reduced free product von Neumann algebra of $M_2(\cc)$ with $M_2(\cc)$. Examples include $AB$ and $A+B$, where A and B are matrices in $(M_2(\cc), {1/2}Tr)*1$ and $1*(M_2(\cc), {1/2}Tr)$, respectively. We prove that AB is an R-diagonal operator (in the sense of Nica and Speicher \cite{N-S1}) 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 \cite{H-S1}, we obtain that if X=AB or X=A+B and $X\neq λ1$, then X has a nontrivial hyperinvariant subspace affiliated with $(M_2(\cc), {1/2}Tr)*(M_2(\cc), {1/2}Tr)$.

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Junsheng Fang, Don Hadwin, Xiujuan Ma. 2007-07-28. On spectra and Brown's spectral measures of elements in free products of matrix algebras. https://arxiv.org/abs/math/0611272

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