arXiv subjects

Sarah L. Browne

Publications and source records attributed to Sarah L. Browne.

7 recordsLinked to original sources

arXiv2026

Functoriality of real crossed product K-theory spectral sequences with respect to group homomorphisms

Spectral sequences are a key tool for computing the K-theory of a crossed product C$^*$-algebra. However, the impact of a group homomorphism $Ω\colon G \to H$ on such a spectral sequence was unknown until quite recently, even when $G = \mathbb Z^\ell$, $H = \mathbb Z^{k}.$ Recent work [Mil25] of the fourth-named author in the complex case establishes that ABC spectral sequences are functorial with respect to group homomorphisms. In this paper, we obtain the analogous result for real K-theory and for united K-theory. Specifically, we first show that the ABC spectral sequence approximates KO$_*(G \ltimes_r A)$ with the group homology H$_p(G;KO_q(A))$ when $G$ is a torsion-free discrete group satisfying the Baum--Connes conjecture with coefficients in $A$. Then, for a homomorphism $Ω\colon G \to H$ of such groups with amenable kernel, and a real $H$-C$^*$-algebra $A$, we show moreover that the map in K-theory induced by the $*$-homomorphism $G \ltimes_r A \to H \ltimes_r A$ is approximated by the natural map in group homology.

math.OA↗

arXiv2023

The Stable Exotic Cuntz Algebras are Higher-Rank Graph Algebras

For each odd integer $n \geq 3$, we construct a rank-3 graph $Λ_n$ with involution $γ_n$ whose real C*-algebra $C^*_\mathbb{R}(Λ_n, γ_n)$ is stably isomorphic to the exotic Cuntz algebra $\mathcal E_n^\mathbb{R}$. This construction is optimal, as we prove that a rank-2 graph with involution $(Λ,γ)$ can never satisfy $C^*_\mathbb{R}(Λ, γ)\sim_{ME} \mathcal E_n^\mathbb{R}$, and the first author reached the same conclusion in previous work. Our construction relies on a rank-1 graph with involution $(Λ, γ)$ whose real C*-algebra $C^*_\mathbb{R}(Λ, γ)$ is stably isomorphic to the suspension $ S \mathbb{R}$. In the Appendix, we show that the i-fold suspension $S^i \mathbb{R}$ is stably isomorphic to a graph algebra iff $-2 \leq i \leq 1$.

math.OA↗

arXiv2022

K-homology and K-theory of pure Braid groups

We produce an explicit description of the K-theory and K-homology of the pure braid group on $n$ strands. We describe the Baum--Connes correspondence between the generators of the left- and right-hand sides for $n=4$. Using functoriality of the assembly map and direct computations, we recover Oyono-Oyono's result on the Baum--Connes conjecture for pure braid groups. We also discuss the case of the full braid group $B_3$.

math.KT↗

arXiv2021

The UCT problem for nuclear $C^\ast$-algebras

In recent years, a large class of nuclear $C^\ast$-algebras have been classified, modulo an assumption on the Universal Coefficient Theorem (UCT). We think this assumption is redundant and propose a strategy for proving it. Indeed, following the original proof of the classification theorem, we propose bridging the gap between reduction theorems and examples. While many such bridges are possible, various approximate ideal structures appear quite promising.

math.OA↗

arXiv2020

E-theory for $C^\ast$-Categories

$E$-theory was originally defined concretely by Connes and Higson and further work followed this construction. We generalise the definition to $C^\ast$-categories. $C^\ast$-categories were formulated to give a theory of operator algebras in a categorical picture and play important role in the study of mathematical physics. In this context, they are analogous to $C^\ast$-algebras and so have invariants defined coming from $C^\ast$-algebra theory but they do not yet have a definition of $E$-theory. Here we define $E$-theory for both complex and real graded $C^\ast$-categories and prove it has similar properties to $E$-theory for $C^\ast$-algebras.

math.OA↗

arXiv2017

A Bott periodicity proof for real graded $C^\ast$-algebras

We give a proof of Bott periodicity for real graded $C^\ast$-algebras in terms of K- theory and E-theory. Guentner and Higson proved a similar result in the complex graded case but we extend this to cover all graded $C^\ast$-algebras. We obtain the 8-fold periodicity in E-theory by constructing two maps that are inverse to each other.

math.KT↗

arXiv2017

E-theory Spectra for graded C*-algebras

This paper brings together C*-algebras and algebraic topology in terms of viewing a C*-algebraic invariant in terms of a topological spectrum. E-theory, E(A,B), is a bivariant functor in the sense that is a cohomology functor in the first variable and a homology functor in the second variable but underlying goes from the category of separable C*-algebras and *-homomorphisms to the category of abelian groups and group homomorphisms. Here we create a generalisation of a orthogonal spectrum to quasi-topological spaces for E-theory. This includes a rich product structure in the context of graded separable C*-algebras.

math.OA↗