$\newcommand{\P}{\operatorname{Proj}}$$\newcommand{\qCP}{\operatorname{CP}}$

Suppose $\theta$ is a projection orthoisomorphism between the set $\qCP(A)$ and $\qCP(B)$ of all q-closed projections of $A=C(X,\mathcal M_m)$ and $B=C(Y,\mathcal M_n)$.

We know that [it preserves 0, order and rank-one projections.][1]

**Theorem 1.** $\theta$ can be extended to a projection orthoisomorphism between

$\ell^\infty(X,\P(A))$ and $\ell^\infty(Y,\P(B))$.

*Proof*.

For each projection $p\in \ell^\infty(X,\P(\mathcal M_m))$, denote by $p_1$ the set of all the rank-one projections majorised by $p$, and define $\tilde\theta(p)\in \ell^\infty(Y, \P(\mathcal M_n))$ by $\tilde\theta(p)=\sup_{m\in p_1}\theta(m)$. This extends $\theta$.

Since any projection in $\mathcal M_m$ is the supremum of rank-one projctions majorised by it,

any projection in $\ell^\infty(X,\P(\mathcal M_m))$ is the supremum of rank-one projctions majorised by it.

From the definition we konw that $\tilde\theta$ preserves order, hence $\{\theta(m)|m\in p_1\}$ consists of all the rank-one projections majorised by $\tilde \theta(p)$. Therefore, $\tilde\theta$ is a bijection.

Since $\tilde\theta$ preserves rank-one projections and order, and a projection $p$ is orthogonal to another projection $q$ in $\ell^\infty(X,\mathcal M_m)$ iff $rs=0$ for all rank-one projection $r$ majorised by $p$ and $s$ majorised by $q$, we have $\tilde\theta$ preserves orthogonality.

By Dye’s theorem, we have

**Corollary.** $\theta$ is implemented by a Jordan *-isomorphism between $\ell^\infty(X,\mathcal M_m)$ and $\ell^\infty(Y,\mathcal M_n)$.

[1]: https://math.liveadvances.com/c-ding/properties-of-a-projection-orthoisomorphism-on-the-set-of-closed-projections/