Vol. 5, No. 1, 2020

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Hausdorffified algebraic $K_1$-groups and invariants for $C^*$-algebras with the ideal property

Guihua Gong, Chunlan Jiang and Liangqing Li

Vol. 5 (2020), No. 1, 43–78
DOI: 10.2140/akt.2020.5.43
Abstract

A C-algebra A is said to have the ideal property if each closed two-sided ideal of A is generated as a closed two-sided ideal by the projections inside the ideal. C-algebras with the ideal property are a generalization and unification of real rank zero C-algebras and unital simple C-algebras. It was long expected that an invariant that we call Inv0(A), consisting of the scaled ordered total K-group (K ¯(A); K ¯(A)+;ΣA)Λ (used in the real rank zero case), along with the tracial state spaces T(pAp) for each cut-down algebra pAp, as part of the Elliott invariant of pAp (for each [p] ΣA), with certain compatibility conditions, is the complete invariant for a certain well behaved class of C-algebras with the ideal property (e.g., AH algebras with no dimension growth). In this paper, we construct two nonisomorphic A𝕋 algebras A and B with the ideal property such that Inv0(A)Inv0(B), disproving this conjecture. The invariant to distinguish the two algebras is the collection of Hausdorffified algebraic K1-groups U(pAp) DU(pAp) ¯ (for each [p] ΣA), along with certain compatibility conditions. We will prove in a separate article that, after adding this new ingredient, the invariant becomes the complete invariant for AH algebras (of no dimension growth) with the ideal property.

Keywords
$C^*$-algebra, $\mathit{AH}$ algebras, ideal property , Elliott invariant, Hausdorffified algebraic $K_1$-group
Mathematical Subject Classification 2010
Primary: 19K14, 19K35, 46L35, 46L80
Milestones
Received: 28 May 2018
Revised: 8 June 2019
Accepted: 24 June 2019
Published: 21 March 2020
Authors
Guihua Gong
College of Mathematics and Information Science
Hebei Normal University
Shijiazhuang, Hebei
China
Department of Mathematics
University of Puerto Rico
Rio Piedras, PR
United States
Chunlan Jiang
College of Mathematics and Information Science
Hebei Normal University
Shijiazhuang, Hebei
China
Liangqing Li
Department of Mathematics
University of Puerto Rico
Rio Piedras, PR
United States