$\boldsymbol{\mathrm{K}_1^Δ}$ for $\boldsymbol{\mathrm{C}^\ast}$-algebras

We introduce a topological abelian group $\mathrm{K}_1^Δ(A)$ for every $\mathrm{C}^\ast$-algebra $A$, which is functorial in $A$ and invariant under stabilisations. $\mathrm{K}_1^Δ$ can distinguish group actions indistinguishable by equivariant (K)K-theory and traces --- we illustrate this on the ir