arXiv subjects

Ellen Weld

Publications and source records attributed to Ellen Weld.

4 recordsLinked to original sources

arXiv2025

Nuclear Dimension and Rigidity Results for Virtually Abelian Groups

Let $G$ be a finitely generated virtually abelian group. We show that the Hirsch length, $h(G)$, is equal to the nuclear dimension of its group $C^*$-algebra, $\dim_{nuc}(C^*(G))$. We then specialize our attention to a generalization of crystallographic groups dubbed \textit{crystal-like}. We demonstrate that in this scenario a \textit{point group} is well defined and the order of this point group is preserved by $C^*$-isomorphism. We close by using these tools to demonstrate that crystallographic (as a group property) is preserved by $C^*$-isomorphism. These three tools combine to prove that $2D$ crystallographic groups are $C^*$-superrigid.

math.OA↗

arXiv2024

The Topology of the Unitary Dual of Crystallography Groups

We provide a procedure for generating the irreducible representations of crystallography groups in any dimension. We also furnish a strategy to investigate the topology of the unitary dual of a crystallography group using sequences of matrices. All irreducible representations (up to unitary equivalence) of the dimension 3 crystallography group 90 and some calculations involving sequences of these irreducible representations are included as a proof of concept of this procedure and strategy.

math.FA↗

arXiv2024

Multiplier algebras of $L^p$-operator algebras

It is known that the multiplier algebra of an approximately unital and nondegenerate $L^p$-operator algebra is again an $L^p$-operator algebra. In this paper we investigate examples that drop both hypotheses. In particular, we show that the multiplier algebra of $T_2^p$, the algebra of strictly upper triangular $2 \times 2$ matrices acting on $\ell_2^p$, is still an $L^p$-operator algebra for any $p$. To contrast this result, we first provide a thorough study of the augmentation ideal of $\ell^1(G)$ for a discrete group $G$. We use this ideal to define a family of nonapproximately unital degenerate $L^p$-operator algebras, $F_{0}^p(\Bbb{Z}/3\Bbb{Z})$, whose multiplier algebras cannot be represented on any $L^q$-space for any $q \in [1, \infty)$ as long as $p \in [1, p_0] \cup [p_0', \infty)$, where $p_0=1.606$ and $p_0'$ is its H\"older conjugate.

math.FA↗

arXiv2019

Connective Bieberbach groups

We prove that a Bieberbach group with trivial center is not connective and use this property to show that a Bieberbach group is connective if and only if it is poly-Z.

math.OA↗