arXiv subjects

Thomas Bray

Publications and source records attributed to Thomas Bray.

3 recordsLinked to original sources

arXiv2026

Minimality for noncommutative dynamical systems

We consider an alternative notion of minimality for C*-dynamical systems, which we refer to as strong minimality. This notion coincides with the usual definition of minimality in the commutative setting, but diverges sharply from it in the noncommutative setting in a way which we argue addresses several fundamental defects in the usual definition. We demonstrate that strong minimality is deeply connected with R{\o}rdam's notion of a C*-irreducible inclusion, giving a new perspective on the study of such inclusions. We provide various characterizations of this property, including for reduced crossed products, tensor products, and subhomogeneous C*-algebras.

math.OA↗

arXiv2026

Simplicity of reduced crossed products

We characterize the simplicity of reduced crossed product C*-algebras in terms of stabilizer subgroups. Specifically, we prove that if $G$ is a countable group and $X$ is a minimal $G$-flow, then the reduced crossed product C*-algebra $\mathrm{C}(X) \times_\lambda G$ is simple if and only if there is a point in $X$ with a C*-simple stabilizer subgroup. Further, these conditions are equivalent to a generic point in $X$ having a C*-simple stabilizer subgroup. We also provide an example demonstrating that this result does not extend to uncountable groups. This completely resolves a question of Ozawa.

math.OA↗

arXiv2025

Statistics on monotonically ordered non-crossing partitions

We study some combinatorial statistics defined on the set $NC^{(mton)}(n)$ of monotonically ordered non-crossing partitions of {1,...,n}, and on the set $NC_2^{(mton)}(2n)$ of monotonically ordered non-crossing pair-partitions of {1,...,2n}. Unlike in the analogous results known for unordered non-crossing partitions, the computations of expectations and variances for natural block-counting statistics on $NC^{(mton)}(n)$ and for the expectation of the area statistic on $NC_2^{(mton)}(2n)$ turn out to yield a logarithmic regime. An important role in our study is played by a nice tree structure on the disjoint union of the $NC^{(mton)}(n)$'s, which we use to streamline our arguments. As an illustration of how these ideas can be applied to calculations of cumulants in monotone probability, we discuss some combinatorial aspects of the monotonic Poisson process.

math.CO↗