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

Establish that for every (real) C*-algebra A, the tautological map on algebraic K-theory K∗(A) → K∗(C([0,1]; A)) induced by the inclusion A → C([0,1]; A) is an isomorphism for all degrees ∗ ≤ 0, i.e., show that the negative algebraic K-groups are invariant under continuous homotopy.

Background

Topological K-theory of C*-algebras is homotopy invariant, but algebraic K-theory is generally not. Rosenberg proposed that, contrary to expectations, negative algebraic K-groups for C*-algebras might still be homotopy invariant. The conjecture has been proven in two extreme cases: commutative C*-algebras (Cortiñas–Thom) and stable C*-algebras (Suslin–Wodzicki). Prior to this work, only K0 or K1 cases were known for arbitrary C*-algebras. This paper proves the first nontrivial case (degree −1) for a broader class of Banach rings, leaving the full range ∗ ≤ 0 open for general C*-algebras.

The authors also show that certain continuity properties fail for lower K-groups of commutative complex Banach algebras, but note that these counterexamples do not contradict Rosenberg’s conjecture, which concerns C*-algebras.