A Note on Positive Zero Divisors in C* Algebras

TL;DR

This paper explores positive zero divisors in C* algebras, introduces a hereditary invariant and zero divisor real rank, and analyzes the structure of graphs formed by these divisors, revealing connectivity and diameter properties.

Contribution

It introduces a hereditary invariant based on zero divisors, defines a new zero divisor real rank, and studies the graph structure of positive zero divisors in C* algebras, including specific cases like the Calkin algebra.

Findings

Hereditary invariant distinguishes certain C* algebras.

Zero divisor real rank is zero for C(X) with specific X.

Graph of positive zero divisors in the Calkin algebra is connected with diameter 3.

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 $\Gamma^{+} (A)$ of non zero positive zero divisors. For the Calkin algebra $A=B(H)/K(H)$, we show that $\Gamma^{+}(A)$ is a connected graph and diam $\Gamma^{+}(A)= 3$. We show that $\Gamma^{+}(A)$ is a connected graph with $\text {diam}\; \Gamma^{+}(A)\leq 6$, if $A$ is a factor.