Random C*-subalgebras
Abstract
Given a discrete countable group Γ, we introduce the concept of random subalgebras of the reduced group C*-algebra Cr*(Γ). This extends the existing theory of invariant random subgroups and the one of invariant random subalgebras of the group von Neumann algebra L(Γ). As applications, we provide a characterisation of C*-simplicity, a condition for amenability of the group and show the existence and uniqueness of a maximal nuclear Γ-invariant unital C*-subalgebra of C*r(Γ). Moreover, we give a topological characterisation of exactness of any separable unital C*-algebra. Finally, we study exotic random subalgebras of C*r(Γ) and super-exotic random subalgebras of L(Γ), that is, random subalgebras which do not arise purely from random subgroups of Γ and not from random subalgebras of C*r(Γ), respectively.
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BibTeX
@misc{fima2026random,
title = {{Random $C^*$-subalgebras}},
author = {Pierre Fima and Hanna Oppelmayer and Leyan Tao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28295},
eprint = {2610.28295},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28295},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Random C^*-subalgebras
AU - Pierre Fima
AU - Hanna Oppelmayer
AU - Leyan Tao
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28295
UR - https://arxiv.org/abs/2610.28295
AB - Given a discrete countable group Γ, we introduce the concept of random subalgebras of the reduced group C^*-algebra C_r^*(Γ). This extends the existing theory of invariant random subgroups and the one of invariant random subalgebras of the group von Neumann algebra L(Γ). As applications, we provide a characterisation of C^*-simplicity, a condition for amenability of the group and show the existence and uniqueness of a maximal nuclear Γ-invariant unital C^*-subalgebra of C^*_r(Γ). Moreover, we give a topological characterisation of exactness of any separable unital C^*-algebra. Finally, we study exotic random subalgebras of C^*_r(Γ) and super-exotic random subalgebras of L(Γ), that is, random subalgebras which do not arise purely from random subgroups of Γ and not from random subalgebras of C^*_r(Γ), respectively.
ER -