Joint spectrum shrinking maps on projections

Let $\mathcal H$ be a finite dimensional complex Hilbert space with dimension $n \ge 3$ and $\mathcal P(\mathcal H)$ the set of projections on $\mathcal H$. Let $\varphi: \mathcal P(\mathcal H) \to \mathcal P(\mathcal H)$ be a surjective map. We show that $\varphi$ shrinks the joint spectrum of any two projections if and only if it is joint spectrum preserving for any two projections and thus is induced by a ring automorphism on $\mathbb C$ in a particular way. In addition, for an arbitrary $k \ge 3$, $\varphi$ shrinks the joint spectrum of any $k$ projections if and only if it is induced by a unitary or an anti-unitary. Assume that $\phi$ is a surjective map on the Grassmann space of rank one projections. We show that $\phi$ is joint spectrum preserving for any $n$ rank one projections if and only if it can be extended to a surjective map on $\mathcal P(\mathcal{H})$ which is spectrum preserving for any two projections. Moreover, for any $k >n$, $\phi$ is joint spectrum shrinking for any $k$ rank one projections if and only if it is induced by a unitary or an anti-unitary.
Comment: 14 pages