- Let $A$ be a simple $C^{*}$ algebra. Assume that the space of projections has a connected component homeomorphic to the complex Grassmanian $G(k,n)$. Is it true to say that, for all $k'<k$, the space of projections has a connected component homeomorphic to $G(k',n)$?
This is motivated by the situation of matrix algebra.
2.There is a natural homeomorphism between projections of $M_{2}(\mathbb{C})$ and $S^{2}$, as a metric subspace of $\mathbb{R}^{3}$, which sends $1/2 \begin{pmatrix} 1-z&x+yi \\x-yi&1+z\end{pmatrix}$ to $(x,y,z)$. Note that this homeomorphism is an isometry, up to a constant. This is a motivation to ask the following two questions:
Q1: Is the space of projections in $M_{2}(\mathbb{C}$) a submanifold with constant sectional curvature.(Motivated by the above isometry and the fact that the sphere has constant curvature). We equip $M_{2}(\mathbb{C})$ with the natural reimannian metric)
Q2: Is there a canonical isometry between projections of matrix algebra with trace $k$ and $G(k,n)$ (with their natural metrics)?
