Rosenberg's conjecture on topological and algebraic K-theory of C*-algebras

Let $A$ be a real C*-algebra. For an abelian group of coefficients, write $K_{*}(A;\mathbf{Z}/n)$ for algebraic K-theory with coefficients and $K^{\textnormal{top}}_{*}(A;\mathbf{Z}/n)$ for topological K-theory with coefficients.

Rosenberg's conjecture.

1. For $n>0$, the map

$$K_{*}(A;\mathbf{Z}/n)\to K^{\textnormal{top}}_{*}(A;\mathbf{Z}/n)$$

is an isomorphism for ${*}\geq0$. 2. The tautological map

$$K_{*}(A)\to K_{*}(\mathscr{C}([0,1];A))$$

is an isomorphism for ${*}\leq0$.

This conjecture compares algebraic and topological K-theory for real C*-algebras and predicts homotopy invariance in the nonpositive degrees. The source presents it as a conjecture of Rosenberg; no resolution is supplied here.