Completely positive coordinates for KK-theory and relative uniform stability
Description
Let A be a unital nuclear C*-algebra, M a von Neumann algebra, and I a norm-closed ideal of M. We prove that a unital completely positive map Φ:A→M, multiplicative modulo I, admits a unital homomorphic correction with the same quotient map whenever
g = sup_{‖a‖≤1} ‖Φ(a*a) − Φ(a)*Φ(a)‖ < 1/2.
The correction has completely bounded error at most min{2, 2g + 2(1 − √(1 − 2g))} ≤ 6g. An amenable-group version uses only the mean unitarity defect of a normalized positive-definite map. Applications give same-size matrix corrections and uniform defect barriers for Bott and index-invisible extension classes. We also give explicit descriptions of both degrees of KK-theory and the Kasparov product through finite-dimensional completely positive systems, and exact fixed-scalar-coefficient UCT reformulations. These descriptions retain independent source-stage and time thresholds and uniform homotopy control; they do not settle the UCT. Finally, the quasimorphism method of Burger–Ozawa–Thom gives almost representations uniformly separated from all positive-definite maps for groups containing a subgroup with noninjective real bounded-cohomology comparison map, including groups containing a nonabelian free group. Assuming the signed-frame input stated below, the same qualitative obstruction extends to every nonamenable discrete group. These results address the positive-definite approximation questions of Pichot and Séguin.