Rosenberg's conjecture for the first negative $K$-group

TL;DR

This paper proves that the negative K-group $K_{-1}$ of Banach rings is invariant under continuous homotopy, countering Rosenberg's conjecture, and shows that similar invariance does not extend to lower K-groups.

Contribution

It establishes the invariance of $K_{-1}$ for Banach rings and provides counterexamples for lower K-groups, clarifying the scope of Rosenberg's conjecture.

Findings

Invariance of $K_{-1}$ under homotopy for Banach rings

Counterexamples for invariance of lower K-groups

Disproof of Rosenberg's conjecture for $K_{-1}$

Abstract

Based on his claims in 1990, Rosenberg conjectured in 1997 that the negative algebraic $K$-groups of C*-algebras are invariant under continuous homotopy. Contrary to his expectation, we prove that such invariance holds for $K_{-1}$ of arbitrary Banach rings by establishing a certain continuity result. We also construct examples demonstrating that similar continuity results do not hold for lower $K$-groups.