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 $AB$ 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 $X\neq \lambda 1$, 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

Effect of sea quarks on the single-spin asymmetries ALW±A^{W^{\pm}}_{L} in polarized pp collisions at RHIC

We calculate the single-spin asymmetries $A^{W^{\pm}}_{L}$ of $W^{\pm}$ bosons produced in polarized pp collisions with the valence part of the up and down quark helicity distributions modeled by the light-cone quark-spectator-diquark model while the sea part helicity distributions of the up and down quarks treated as parametrization. Comparing our results with those from experimental data at RHIC, we find that the helicity distributions of sea quarks play an important role in the determination of the shapes of $A^{W^{\pm}}_{L}$. It is shown that $A^{W^{-}}_{L}$ is sensitive to $\Delta \bar u$, while $A^{W^{+}}_{L}$ to $\Delta \bar d$ intuitively. The experimental data of the polarized structure functions and the sum of helicities are also important to constrain the sizes of quark helicity distributions both for the sea part and the valence part of the nucleon.Comment: 19 latex pages, 5 figures, final version for publicatio