Spectrum of reduced von Neumann algebra

Question

Let $M$ be a von Neumann algebra and $e$ a projector. Do you know if $Spec_{eMe}(exe)\subset Spec_M(x)$ for $x\in M$? Thank you

Answer 1

No. Given any positive contraction $c$, the $2\times 2$ matrix $$ \begin{bmatrix} c& (c-c^2)^{1/2}\\ (c-c^2)^{1/2}&1-c\end{bmatrix} $$ is a projection. This allows you to construct $x$ and $e$ such that $\sigma_M(x)=\{0,1\}$ but $\sigma_{eMe}(exe)$ can be any closed subset of the unit interval.

Similarly, if $v$ is a contraction, $$ \begin{bmatrix} v& (1-vv^*)^{1/2}\\ (1-v^*v)^{1/2}&v^*\end{bmatrix} $$ is a unitary. So in this case $\sigma_M(x)$ is in the unit circle, while $\sigma_{eMe}(exe)$ can be any closed subset of the closed unit disk.