On the set of orthogonal positive sums in a $C^*$-algebra

Abstract

For a $C^*$-algebra $A$ let $\mathcal I(A)$ be the set of all elements $a+b$ with $a,b\in A_+$, $ab=0$ and $\|a\|=\|b\|=1$. We study the algebraic, metric and topological properties of $\mathcal I(A)$. We describe $\mathcal I(A\oplus B)$ and show that $\mathcal I(A/J)\subseteqπ(\mathcal I(A))$ for the quotient map $π:A\to A/J$. For finite-dimensional $A$ we prove $\mathcal I(A)=\operatorname{Sph}^+_A(E(A))$, where $E(A)$ is the set of minimal projections of $A$ and $\operatorname{Sph}^+_A(E)$ 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 $\mathcal I(A)\neq\varnothing$ 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 $\mathcal I(\ell_c(A))$ fails to be closed when $A$ is unital without nontrivial projections.