MathOverflow

I have a question regarding the argument in the answer by user491858 to my earlier question. Please visit the original question at first. Apparently, the answer seems very insightful, however, I would appreciate some clarification on the following points...

Let $M$ be a von Neumann algebra and $M_*$ be its predual. Then $M_*$ is 1-Plichko [1].
Q: Does there exist a (commutative) 1-projectional skeleton $\{P_s:s\in J\}$ on $M_*$ such that each $P_s^*:M\to M$ is a conditional expectation?
If so, each $P_s$ must be completely contractive and satisfy $P_s^*(1) = 1$....

Let $\Lambda$ be the ring of symmetric functions and let $\phi:\Lambda \to \mathbb{C}$ be a Schur-positive functional such that $\phi(s_\lambda)\in \mathbb{Z}_{\ge 0}$ for all partitions $\lambda$ (for instance arising from a rational totally positive...

Delzant’s theorem classifies compact symplectic toric manifolds via Delzant polytopes. This can be viewed functorially.
Consider the directed system of simplices $\Delta^n \subset \mathbb{R}^{n+1}$ and projective spaces $\mathbb{CP}^n$. Their colimits yield, respectively, a countable simplex and $\mathbb{CP}^\infty$. Both admit natural completions: for instance,$$\Delta...