D-Bar Operators on Quantum Domains

In this paper we analyse for a $G$-$C^{*}$-algebra $A$ to which extent one can calculate the $K$-theory of the reduced crossed product $K(A\rtimes_{r}G)$ from the $K$-theory spectrum $K(A)$ with the induced $G$-action. We also consider some cases where one allows to use the $K$-theories of crossed products for some proper subgroups of $G$. Our central goal is to demonstrate the usefulness of a homotopy theoretic approach. We mainly concentrate on finite groups.

math.OA↗

arXiv2026

A Hilbert Module Approach to Classical and Quantum Reference Frames

In this paper, we present a model for classical and quantum reference frames for $C^*$-algebras by utilizing Hilbert modules and equivariant correspondences of $C^*$-dynamical systems. We show the classical model provides a commutative baseline for the quantum model, and we discuss how the quantum model encodes the Hilbert space operational formulation of quantum reference frames. We also include various explicit examples for both classical and quantum reference frames.

math.OA↗

arXiv2026

Single commutators in $C^*$-algebras

Let $\mathcal{A}$ be a simple unital $C^*$-algebra with real rank zero, nonempty tracial state space $T(\mathcal{A})$, and strict comparison of projections. We prove that every element $T$ in $\mathcal{A}$ with $τ(T)=0$ for all $τ\in T(\mathcal{A})$ is a single commutator if one of the following conditions holds: $(1)$ $\mathcal{A}$ has unique trace; $(2)$ $\mathcal{A}$ is separable and has stable rank one. We also prove that every nonscalar element in a simple unital purely infinite $C^*$-algebra is a single commutator. Applications include UHF algebras, irrational rotation algebras, and Cuntz algebras.

math.OA↗