# Journal of Operator Theory

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

Comparisons of equivalence relations on open projections

Authors:  Chi-Keung Ng (1) and Ngai-Ching Wong (2)
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) and the PZ-equivalence (as studied by C.~Peligrad and L. Zsid\'{o} as well as by E. Ortega, M. R{\o}rdam 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, R{\o}rdam and Thiel) coincide on open projections of $C_0(\Omega)\otimes \CK(\ell^2)$ exactly when the canonical homomorphism from $\Cu(C_0(\Omega))$ into $\lsc(\Omega;\overline \BN_0)$ is bijective. Here, $\Cu(C_0(\Omega))$ is the stabilized Cuntz semigroup, and $\lsc(\Omega;\overline \BN_0)$ is the semigroup of lower semicontinuous functions from $\Omega$ into $\overline{\BN}_0 := \{0,1,2,\ldots, \infty\}$.

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