arXiv subjects

Connor MacMahon

Publications and source records attributed to Connor MacMahon.

4 recordsLinked to original sources

arXiv2026

1-Bounded Entropy for $C^*$-Algebras

Towards a question of Voiculescu, two notions of $1$-bounded entropy, $h$ and $h^\text{top}$, are defined for $C^*$-algebras. The quantity $h$ involves the tracial completion with respect to all traces while the quantity $h^\text{top}$ depends on operator norm microstates. It is demonstrated that $h^\text{top}(\mathscr{A})$ does not exceed $h(\mathscr{A})$ for all $C^*$-algebras $\mathscr{A}$. Moreover, a variational principle enabling the computation of $h$ is introduced. The quantity $h$ is computed in various examples, and the computation is used to justify structural conclusions such as $C^*$-primeness, crossed product indecomposability, and free indecomposability. These notions of entropy are generalized to operator systems, where it is demonstrated that both may be computed on a basis.

math.OA↗

arXiv2026

Existential Inclusions of Bi-exact Groups are Conjugacy Representation Rigid

If $\Lambda$ is a non-amenable bi-exact group and $\Lambda \hookrightarrow \Gamma$ is an existential embedding, then each of the intersections $\Lambda \cap g \Lambda g^{-1}$ for $g$ a member of $\Gamma \backslash \Lambda$ is amenable. This in conjunction with work of Bekka and Kalantar demonstrates that in this situation, the weak equivalence class of the quasi-regular representation $\lambda_{\Gamma/\Lambda}$ determines $\Lambda$ up to conjugacy among the self-commensurating subgroups of $\Gamma$.

math.GR↗

arXiv2026

An $\ell^2$ Obstruction for Elementary Embeddings of Hyperbolic Groups

The first $\ell^2$ Betti number of a group is non-decreasing under various embeddings arising from first order logic. Strict inequality is proved for elementary embeddings of non-abelian proper subgroups within torsion free hyperbolic groups using Perin's classification of such inclusions. The monotonicity is further demonstrated for existential embeddings of arbitrary finitely generated groups.

math.GR↗

arXiv2022

Large sets avoiding infinite arithmetic / geometric progressions

We study some variants of the Erd\H{o}s similarity problem. We pose the question if every measurable subset of the real line with positive measure contains a similar copy of an infinite geometric progression. We construct a compact subset $E$ of the real line such that $0$ is a Lebesgue density point of $E$, but $E$ does not contain any (non-constant) infinite geometric progression. We give a sufficient density type condition that guarantees that a set contains an infinite geometric progression. By slightly improving a recent result of Bradford, Kohut and Mooroogen arXiv:2205.04786, we construct a closed set $F\subset[0,\infty)$ such that the measure of $F\cap[t,t+1]$ tends to $1$ at infinity but $F$ does not contain any infinite arithmetic progression. We also slightly improve a more general recent result by Kolountzakis and Papageorgiou arXiv:2208.02637 for more general sequences. We give a sufficient condition that guarantees that a given Cantor type set contains at least one infinite geometric progression with any quotient between $0$ and $1$. This can be applied to most symmetric Cantor sets of positive measure.

math.MG↗