Algebraic K-theory, K-regularity, and T-duality of O∞-stable C*-algebras

We develop an algebraic formalism for topological T-duality. More precisely, we show that topological T-duality actually induces an isomorphism between noncommutative motives that in turn implements the well-known isomorphism between twisted K-theories (up to a shift). In order to establish this result we model topological K-theory by algebraic K-theory. We also construct an E-infinity-operad starting from any strongly self-absorbing C*-algebra D. Then we show that there is a functorial topological K-theory symmetric spectrum construction K-Sigma(top) (-) on the category of separable C*-algebras, such that K-Sigma(top) (D) is an algebra over this operad; moreover, K-Sigma(top) (A (circle times) over cap D) is a module over this algebra. Along the way we obtain a new symmetric spectra valued functorial model for the (connective) topological K-theory of C*-algebras. We also show that O-infinity-stable C*-algebras are K-regular providing evidence for a conjecture of Rosenberg. We conclude with an explicit description of the algebraic K-theory of ax + b-semigroup C*-algebras coming from number theory and that of O-infinity-stabilized noncommutative tori.