Uncountable Abelian Group C*-algebras Fail the Lifting Property

Miles Gould

arXiv:2607.20407v1 Primary: math.OA

Abstract

In this note, we show that $C^*(G)$ fails the lifting property (LP) for every discrete group $G$ containing an uncountable abelian subgroup. Consequently, for every uncountable discrete abelian group $G$, the nuclear algebra $C^*(G)$ has the local lifting property (LLP) but fails the LP. This shows that the LP and the LLP do not coincide for full discrete group $C^*$-algebras.

Disclosure

“cular, after attending Koszmider’s talk on the contents of [8], Willett suggested to me that the results therein could be of use. This insight was precisely the key to finding the first example, so I am particularly grateful to him for it. ChatGPT was used substantially in the writing of this note, both in mathematical content and proofreading. I wrote and checked the final version of every proof appearing, and assume responsibility for their validity.”
PDF page 2

Classification: Proof ideas or individual proof-step assistance
Multiplier: 8
Verified: 2026-08-23T19:02:06.009783Z

Structural counts

Pages 7 pdf

Theorems 5 source

Lemmas 5 source

Propositions 0 source

Corollaries 0 source

Definitions 1 source

Displayed equations 22 source

Bibliography entries 0 source

Appendix pages 0 estimated

Count notes

Source counts use the expanded primary TeX file UAGCFtLP__2_.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.