A Note on Positive Zero Divisors in C* Algebras

Abstract

In this paper we concern with positive zero divisors in $C^{*}$ algebras. By means of zero divisors, we introduce a hereditary invariant for $C^{*}$ algebras. Using this invariant, we give an example of a $C^{*}$ algebra $A$ and a $C^{*}$ sub algebra $B$ of $A$ such that there is no a hereditary imbedding of $B$ into $A$. We also introduce a new concept zero divisor real rank of a $C^{*}$ algebra, as a zero divisor analogy of real rank theory of $C^{*}$ algebras. We observe that this quantity is zero for $A=C(X)$ when $X$ is a separable compact Hausdorff space or $X$ is homeomorphic to the unit square with the lexicographic topology. To a $C^{*}$ algebra $A$ with $\dim A > 1$, we assign the undirected graph $Γ^{+} (A)$ of non zero positive zero divisors. For the Calkin algebra $A=B(H)/K(H)$, we show that $Γ^{+}(A)$ is a connected graph and diam $Γ^{+}(A)= 3$. We show that $Γ^{+}(A)$ is a connected graph with $\text {diam}\; Γ^{+}(A)\leq 6$, if $A$ is a factor.