On the set of orthogonal positive sums in a $C^{\ast }$-algebra

For a $C^{\ast }$-algebra $A$ let $\mathcal{I}(A)$ be the set of all elements $a+b$ with $a,b\in A_{+}$, $ab=0$ and $\mathrm{\parallel }a\mathrm{\parallel }=\mathrm{\parallel }b\mathrm{\parallel }=1$. We study the algebraic, metric and topological properties of $\mathcal{I}(A)$. We describe $\mathcal{I}(A\oplus B)$ and show that $\mathcal{I}(A\mathrm{/}J)\subseteq \pi (\mathcal{I}(A))$ for the quotient map $\pi :A\to A\mathrm{/}J$. For finite-dimensional $A$ we prove $\mathcal{I}(A)={\mathrm{Sph}}_A^{+}(E(A))$, where $E(A)$ is the set of minimal projections of $A$ and ${\mathrm{Sph}}_A^{+}(E)$ is the set of positive norm-one elements at distance one from every element of $E$; we also show that the analogous statement with the minimal projections of the bidual fails in general. When $\mathcal{I}(A)\mathrm{\ne }\mathrm{\varnothing }$ the distance from $1$ to $\mathcal{I}(A)$ is $0$ or $1$, according to whether $A$ has a nontrivial projection. Moreover, $\mathcal{I}(A)$ is norm compact if and only if $A$ is finite-dimensional, closedness of $\mathcal{I}(A)$ passes to hereditary subalgebras, $\mathcal{I}(A)$ is norm closed for every von Neumann algebra, and the extreme points of $\mathcal{I}(A)$ are exactly its projections. For a compact, locally path connected space $X$ the set $\mathcal{I}(C(X))$ is closed, whereas $\mathcal{I}({\mathrm{\ell }}_c(A))$ fails to be closed when $A$ is unital without nontrivial projections.