OAI.WeakPureInfiniteness.exact_weaklyPurelyInfinite_individuallyProperlyInfinite_any

The theorem states that, for a non-unital C*-algebra A (with its partial order making it star-ordered) that is Exact and WeaklyPurelyInfinite, every positive element a (0 ≤ a, with no nonzero assumption) has a properly infinite one-by-one matrix over A. Here Exact A means: for every surjective ∗-homomorphism q : C → D between non-unital C*-algebras in the same universe as A, any element z of the minimal (spatial) tensor product of A and C whose image vanishes under every spatial representation of A and D pulled back along q (that is, evaluated at the representation with right factor precomposed with q) already lies in the closed star-subalgebra generated by the elementary tensors a ⊗ c with q(c) = 0. WeaklyPurelyInfinite A means there is some n ≥ 1 such that, for every nonzero positive a in A, the n-fold diagonal matrix diag(a, …, a) is properly infinite. A matrix x is properly infinite when the block-diagonal matrix diag(x, x) is Cuntz-subequivalent to x, meaning that for every ε > 0 there is a rectangular matrix v with ‖v*·x·v − diag(x, x)‖ < ε. The conclusion is this property for the one-copy diagonal matrix of a, which is just a viewed as a 1×1 matrix, so the n in the hypothesis is replaced by 1 and zero is included.