Weak pure infiniteness and Cuntz-algebra absorption

The explanation

Take the space of infinite sequences with basis vectors e0, e1, e2, and so on. Let S1 send e_n to e_(2n) and S2 send e_n to e_(2n+1). So S1 sends e3 to e6, and S2 sends e3 to e7. Each map keeps lengths. S1 fills the even slots and S2 fills the odd slots, with no overlap. So the identity holds two full copies of itself side by side. Matrices cannot do this: two copies of the k-by-k identity have trace 2k, more than k.

A C*-algebra is a family of operators closed under sums, products, adjoints and limits. A positive element h is properly infinite when h ⊕ h fits below h in the sense of Cuntz comparisonSource 1. Cuntz comparison means h ⊕ h can be squeezed into h by operators, up to small errors. Pure infiniteness asks this for every positive element. Weak pure infiniteness asks it only after taking n copies of each element, for one fixed n. Strong pure infiniteness also requires two elements to be cut apart at once, with their mixed entry made smallSource 2. Strong implies ordinary, and ordinary implies weak. Kirchberg and Rørdam asked whether the reverse implications holdSource 3. The obstacle is the mixed entry. Knowing each element alone is infinite says nothing direct about how two elements interact.

The paper proves that if every positive element is properly infinite, the algebra is strongly purely infiniteSource 4. It needs no exactness, nuclearity, unitality or simplicity. For exact algebras, it proves that the fixed-amplification property already makes every positive element properly infiniteSource 5. Exact algebras are a large, well-behaved class defined through tensor products. So all three notions agree for exact algebrasSource 6. Combined with earlier results of Kirchberg and Rørdam, every separable nuclear algebra with the fixed-amplification property absorbs O∞Source 7. Absorbing O∞ means A ⊗ O∞ is again A, where O∞ is generated by infinitely many isometries like S1 and S2.

How sure is this? Lean, a computer proof checker, has formal statements of the two main theorems. They are the ordinary-to-strong implication and the fixed-amplification result for exact algebras. So the paper proves both, as far as the Lean statements match the paper. The O∞ absorption step uses earlier published results. No specialist review is published.

What it doesn't show. This does not show that weak pure infiniteness implies ordinary pure infiniteness for algebras that are not exact.