1,675 research outputs found
On maximal injective subalgebras of tensor products of von Neumann algebras
Let M_i be a von Neumann algebra, and B_i be a maximal injective von Neumann
subalgebra of M_i, i=1,2. If M_1 has separable predual and the center of B_1 is
atomic, e.g., B_1 is a factor, then B_1\tensor B_2 is a maximal injective von
Neuamnn subalgebra of M_1\tensor M_2. This partly answers a question of PopaComment: 15 pages, a simple proof of the main theorem (Theorem 4.1 in the new
version) suggested by referee is included in the new version. Typos are
correcte
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
(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 and , 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 , then X has
a nontrivial hyperinvariant subspace affiliated with (M_2(\cc),
{1/2}Tr)*(M_2(\cc), {1/2}Tr).Comment: final version. to appear on Math. Sca
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