${\mathrm{K}}_1^{\mathbf{\Delta }}$ for ${\mathrm{C}}^{\ast }$-algebras

Abstract

We introduce a topological abelian group ${\mathrm{K}}_1^{\Delta }(A)$ for every ${\mathrm{C}}^{\ast }$-algebra $A$, which is functorial in $A$ and invariant under stabilisations. ${\mathrm{K}}_1^{\Delta }$ can distinguish group actions indistinguishable by equivariant (K)K-theory and traces --- we illustrate this on the irrational rotation algebra by exhibiting strongly outer actions of $\mathbb{Z}$ which, via ${\mathrm{K}}_1^{\Delta }$, are not (stably) cocycle conjugate. These actions are moreover trivial in ${\mathrm{KK}}^{\mathbb{Z}}$ and on the Hausdorffised unitary algebraic ${\mathrm{K}}_1$ group ${\overline{\mathrm{K}}}_1^{\mathrm{alg}}(A)$. The Hausdorffisation ${\overline{\mathrm{K}}}_1^{\Delta }(A)$ of ${\mathrm{K}}_1^{\Delta }(A)$ forms the final part of the total stable invariant ${\underline{\mathrm{K}}\mathrm{T}}_{\mathrm{s}}(A)$, which we also introduce. We prove that ${\underline{\mathrm{K}}\mathrm{T}}_{\mathrm{s}}$ classifies $\ast$-homomorphisms up to approximate unitary equivalence between certain stable ${\mathrm{C}}^{\ast }$-algebras which contain a non-zero projection. ${\mathrm{K}}_1^{\Delta }(A)$ is new even in the unital setting, but under further assumptions on $A$ recovers the unitary algebraic ${\mathrm{K}}_1$ group ${\mathrm{K}}_1^{\mathrm{alg}}(A)$, whereas the Hausdorffisations ${\overline{\mathrm{K}}}_1^{\Delta }(A)$ and ${\overline{\mathrm{K}}}_1^{\mathrm{alg}}(A)$ agree for every unital $A$.

arXiv comment47 pages