Journal of Operator Theory

Volume 74, Issue 1, Summer 2015 pp. 101-123.

Comparisons of equivalence relations on open projections

Authors: (1) Chi-Keung Ng, (2) Ngai-Ching Wong
Author institution: (1) Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China
(2) Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, 80424, Taiwan

Summary: The aim of this article is to compare some equivalence relations among open projections of a $C^*$-algebra. Such equivalences are crucial in a decomposition scheme of $C^*$-algebras and is related to the Cuntz semigroups of $C^* $-algebras. In particular, we show that the spatial equivalence (as studied by H. Lin as well as by the authors)i and the PZ-equivalence (as studied by C. Peligrad and L. Zsidó as well as by E. Ortega, M. Rørdam and H. Thiel) are different, although they look very similar and conceptually the same. In the development, we also show that the Murray--von Neumann equivalence and the Cuntz equivalence (as defined by Ortega, Rørdam and Thiel) coincide on open projections of $C_0(\Omega)\otimes \mathcal K(\ell^2)$ exactly when the canonical homomorphism from $\mathrm{Cu}(C_0(\Omega))$ into $\mathrm{Lsc}(\Omega;\overline {\mathbb{N}}_0)$ is bijective. Here, $\mathrm{Cu}(C_0(\Omega))$ is the stabilized Cuntz semigroup, and $\mathrm{Lsc} (\Omega;\overline {\mathbb{N}}_0)$ is the semigroup of lower semicontinuous functions from $\Omega$ into $\overline{\mathbb{N}}_0 := \{0,1,2,\ldots, \infty\}$.

DOI: http://dx.doi.org/10.7900/jot.2014may06.2045
Keywords: $C^*$-algebra, open projection, equivalence relation, Cuntz semigroup

Contents Full-Text PDF