A Note on Positive Zero Divisors in C* Algebras

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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 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 diam Γ⁺(A)≤ 6, if A is a factor.