C^*-extreme maps and *-homomorphisms from C(X) to finite von Neumann algebras

@misc{giannakis2026cextreme,
  title = {{$C^*$-extreme maps and $*$-homomorphisms from $C(X)$ to finite von Neumann algebras}},
  author = {Dimitrios Giannakis and Michael Montgomery and Travis Russell},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.22685},
  eprint = {2609.22685},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.22685},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - C^*-extreme maps and *-homomorphisms from C(X) to finite von Neumann algebras
AU  - Dimitrios Giannakis
AU  - Michael Montgomery
AU  - Travis Russell
PY  - 2026
DA  - 2026/09/19
PB  - arXiv
DO  - 10.48550/arxiv.2609.22685
UR  - https://arxiv.org/abs/2609.22685
AB  - Given a unital inclusion of C^*-algebras 𝒞 ⊂ 𝒜 and a unital inclusion 𝒞 ⊂ ℬ into a von Neumann algebra ℬ, we investigate the extreme points of unital completely positive maps from 𝒜 to ℬ that fix 𝒞 denoted by UCP_ C(𝒜, ℬ). This space is obviously convex and C^*-convex with respect to the C^*-algebra 𝒞' ∩ ℬ. In this article we show that the 𝒞' ∩ ℬ-extreme points are exactly the *-homomorphisms that fix 𝒞 when 𝒜 is commutative and ℬ has a normal faithful center valued trace. This generalizes a result due to Farenick and Morenz where 𝒞 = ℂ 1 and ℬ = M_n(ℂ).
ER  -