Non-stable K-theory and extremally rich C*-algebras

Lawrence G. Brown; Gert K. Pedersen

Home/Archive/arXiv

arXiv · 0708.3078

Non-stable K-theory and extremally rich C*-algebras

Lawrence G. Brown·Gert K. Pedersen

View original source ↗Read source document ↗Export metadata ↓

Abstract

We consider the properties weak cancellation, K_1-surjectivity, good index theory, and K_1-injectivity for the class of extremally rich C*-algebras, and for the smaller class of isometrically rich C*-algebras. We establish all four properties for isometrically rich C*-algebras and for extremally rich C*-algebras that are either purely infinite or of real rank zero, K_1-injectivity in the real rank zero case following from a prior result of H. Lin. We also show that weak cancellation implies the other properties for extremally rich C*-algebras and that the class of extremally rich C*-algebras with weak cancellation is closed under extensions. Moreover, we consider analogous properties which replace the group K_1(A) with the extremal K-set K_e(A) as well as two versions of K_0-surjectivity.

Explore related subjects

math.OA

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Lawrence G. Brown, Gert K. Pedersen. 2014-04-03. Non-stable K-theory and extremally rich C*-algebras. https://arxiv.org/abs/0708.3078

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The UMD property of symmetric operator spaces

We prove that if $E$ is a UMD symmetric Banach function space on $(0,\infty)$, then $E(\mathcal{M},τ)$ is UMD for every semifinite von Neumann algebra $\mathcal{M}$ equipped with a faithful normal semifinite trace $τ$. This resolves in the affirmative an open problem that has circulated in the non-commutative world for more than four decades.

math.OA↗

arXiv2026

Revisiting the Transfinite Christensen-Pedersen Argument

Christensen and Pedersen proved that every properly infinite $\mathrm{AW}^*$-algebra is monotone sequentially complete, and Saitô and Wright developed a transfinite form of their dilation argument. We revisit the transfinite construction using normality of $\mathrm{AW}^*$-algebras. Normality simplifies the limit stages by turning suprema into compressions of joins, so the construction only needs a supply of fresh orthogonal projections large enough to contain the supports of the summands at successor stages. We use this simplified proof to show that a $*$-homomorphism between $\mathrm{AW}^*$-algebras that preserves only the joins needed to encode such a sum preserves the sum itself. We also use it to deduce order-continuity facts about $κ$-join-preserving $*$-homomorphisms. We also show that a finite $\mathrm{AW}^*$-algebra has suprema for all bounded positive families whose supports have bounded total center-valued dimension.

math.OA↗

arXiv2026

Vertex-transitive quantum graphs

We define a quantum graph to be vertex-transitive if the join of its automorphism group is the maximum quantum relation on its quantum vertex set, in direct analogy with the classical case. All simple quantum graphs in $M_2(\mathbb C)$ are vertex-transitive, but many simple quantum graphs in $M_3(\mathbb C)$ are not vertex-transitive. We provide a complete classification of vertex-transitive quantum graphs in $M_3(\mathbb C)$ up to isomorphism. To do this, we introduce a polynomial invariant for quantum graphs in $M_n(\mathbb C)$, which we call the panoramic polynomial.

math.OA↗