K1Δ for C∗-algebras
Abstract
We introduce a topological abelian group K1Δ(A) for every C∗-algebra A, which is functorial in A and invariant under stabilisations. K1Δ 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 ℤ which, via K1Δ, are not (stably) cocycle conjugate. These actions are moreover trivial in KKℤ and on the Hausdorffised unitary algebraic K1 group K̅1alg(A). The Hausdorffisation K̅1Δ(A) of K1Δ(A) forms the final part of the total stable invariant underlineKTs(A), which we also introduce. We prove that underlineKTs classifies *-homomorphisms up to approximate unitary equivalence between certain stable C∗-algebras which contain a non-zero projection. K1Δ(A) is new even in the unital setting, but under further assumptions on A recovers the unitary algebraic K1 group K1alg(A), whereas the Hausdorffisations K̅1Δ(A) and K̅1alg(A) agree for every unital A.
BibTeX
@misc{gabe2026k1δ,
title = {{$\boldsymbol\mathrmK_1^Δ$ for $\boldsymbol\mathrmC^\ast$-algebras}},
author = {James Gabe and Alistair Miller},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06187},
eprint = {2610.06187},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06187},
}
RIS
TY - UNPB
TI - K_1^Δ for C^∗-algebras
AU - James Gabe
AU - Alistair Miller
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06187
UR - https://arxiv.org/abs/2610.06187
AB - We introduce a topological abelian group K_1^Δ(A) for every C^∗-algebra A, which is functorial in A and invariant under stabilisations. K_1^Δ 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 ℤ which, via K_1^Δ, are not (stably) cocycle conjugate. These actions are moreover trivial in KK^ℤ and on the Hausdorffised unitary algebraic K_1 group K̅_1^alg(A). The Hausdorffisation K̅_1^Δ(A) of K_1^Δ(A) forms the final part of the total stable invariant underlineKT_s(A), which we also introduce. We prove that underlineKT_s classifies *-homomorphisms up to approximate unitary equivalence between certain stable C^∗-algebras which contain a non-zero projection. K_1^Δ(A) is new even in the unital setting, but under further assumptions on A recovers the unitary algebraic K_1 group K_1^alg(A), whereas the Hausdorffisations K̅_1^Δ(A) and K̅_1^alg(A) agree for every unital A.
ER -