OAI.WeakPureInfiniteness.exact_weaklyPurelyInfinite_stronglyPurelyInfinite

The theorem states that for every non-unital C*-algebra A (with a partial order making it a star-ordered ring), if A is exact and weakly purely infinite, then A is strongly purely infinite. Exactness of A means: for all non-unital C*-algebras C and D in the same universe and every surjective star-homomorphism q : C → D, any element z of the minimal (spatial) tensor product of A and C whose image vanishes under every spatial representation of A ⊗ D pulled back along q (that is, z evaluates to zero at every representation of A and D on Hilbert spaces, with the D-representation precomposed with q) must lie in the closed star-subalgebra generated by the elementary tensors a ⊗ c with q(c) = 0. Weakly purely infinite means there is an integer n ≥ 1 such that, for every nonzero positive a in A, the n×n diagonal matrix over A with all diagonal entries a is properly infinite, where a matrix x is properly infinite if the block sum x ⊕ x is Cuntz-subequivalent to x, meaning that for every ε > 0 there is a matrix v over A, with rows indexed like x and columns indexed like x ⊕ x, such that ‖vxv − (x ⊕ x)‖ < ε. Strongly purely infinite means that for all positive a, b in A and every ε > 0 there exist s, t in A such that ‖sa²s − a²‖ < ε, ‖tb²t − b²‖ < ε, and ‖sabt‖ < ε.