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A simple C*-algebra which is not K1-injective

  • Andrew S. Toms
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Abstract

We exhibit a simple unital infinite-dimensional C*-algebra A which is not K1-injective, i.e., the canonical map $ U(A)/U_0(A)\longrightarrow K_1(A)$ is not injective. This phenomenon is well known in the non-simple case. Indeed, a natural example is the generator u of π4(U(2))≅ℤ/2, and we use exactly this example as the seed of our construction. The construction proceeds by developing a higher-rank matrix trapping technique in homogeneous algebras generalizing the machinery of Villadsen in higher stable rank simple C*-algebras. A hypothetical homotopy from the homomorphic image of u to the identity in a certain homogeneous algebra is shown, through matrix trapping, to yield a spin bordism from the class of u to that of the identity matrix 12 ∈ M2, and this, in turn, is obstructed by the fact that the bordism class of u is not trivial. We then fit our new arguments into a generalized version of Villadsen's higher stable rank construction to arrive at a simple example.

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Cite

BibTeXRIS
BibTeX
@misc{toms2026simple,
  title = {{A simple C$^*$-algebra which is not $K_1$-injective}},
  author = {Andrew S. Toms},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.09535},
  eprint = {2609.09535},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.09535},
}
RIS
TY  - UNPB
TI  - A simple C^*-algebra which is not K_1-injective
AU  - Andrew S. Toms
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.09535
UR  - https://arxiv.org/abs/2609.09535
AB  - We exhibit a simple unital infinite-dimensional C^*-algebra A which is not K_1-injective, i.e., the canonical map $ U(A)/U_0(A)\longrightarrow K_1(A)$ is not injective. This phenomenon is well known in the non-simple case. Indeed, a natural example is the generator u of π_4(U(2))≅ℤ/2, and we use exactly this example as the seed of our construction. The construction proceeds by developing a higher-rank matrix trapping technique in homogeneous algebras generalizing the machinery of Villadsen in higher stable rank simple C^*-algebras. A hypothetical homotopy from the homomorphic image of u to the identity in a certain homogeneous algebra is shown, through matrix trapping, to yield a spin bordism from the class of u to that of the identity matrix 1_2 ∈ M_2, and this, in turn, is obstructed by the fact that the bordism class of u is not trivial. We then fit our new arguments into a generalized version of Villadsen's higher stable rank construction to arrive at a simple example.
ER  -