Zero-divisor graphs and zero-divisor real rank of $C^{\ast }$-algebras

We study two invariants of $C^{\ast }$-algebras built from positive zero divisors. First, we associate with a $C^{\ast }$-algebra $A$ the graph ${\mathrm{\Gamma }}_{+}(A)$ whose vertices are the nonzero positive zero divisors of $A$, two vertices being adjacent when they are orthogonal. We show that ${\mathrm{\Gamma }}_{+}$ of the Calkin algebra is connected of diameter $3$, that ${\mathrm{\Gamma }}_{+}(B(H))$ is connected of diameter $4$ (for $\dim H\ge 3$), that ${\mathrm{\Gamma }}_{+}(A)$ is connected of diameter at most $4$ for every von Neumann factor $A\mathrm{\ne }\mathbb{C},M_2(\mathbb{C})$, and of diameter at most $3$ if $A$ is not of type~I, whereas ${\mathrm{\Gamma }}_{+}(M_2(\mathbb{C}))$ is disconnected. Second, we study the zero-divisor real rank $ZDRR(A)$, obtained by replacing invertibility in the definition of real rank with the weaker condition of being a non-zero-divisor, so that $ZDRR(A)\le RR(A)$. For commutative algebras $C(X)$ we prove $ZDRR(C(X))=RR(C(X))=\dim X$ when $X$ is a compact almost-$P$-space, while $ZDRR(C(X))=0$ for separable $X$. Using Levy’s connected, totally ordered almost-$P$-space we construct unital commutative $C^{\ast }$-algebras $A_n$ with $ZDRR(A_n)=n$ for every $n\in {\mathbb{N}}_0$. On the other hand $ZDRR(C(X\times [0,1]))=0$ for every compact Hausdorff space $X$, so $ZDRR$ and $RR$ can differ arbitrarily.