Rosenberg’s homotopy invariance conjecture for negative algebraic K-theory of (real) C*-algebras

Establish that for every real C*-algebra A, the tautological map on algebraic K-theory K_{*}(A) → K_{*}(C([0,1]; A)) is an isomorphism for all degrees * ≤ 0, thereby proving homotopy invariance of negative algebraic K-theory along the interval [0,1] for arbitrary (possibly noncommutative) real C*-algebras.

Background

The conjecture originates from Rosenberg’s program connecting algebraic K-theory of operator algebras with topological invariance properties. Cortiñas–Thom proved the commutative complex case, and the present paper settles the commutative real case (see Theorem main_kr), but the full noncommutative (real) C*-algebra case remains unresolved.

The authors indicate plans to reinterpret such statements using condensed mathematics and to provide a proof in degree K_{-1}, highlighting ongoing progress while underscoring that the complete conjecture across all negative degrees remains an open problem.