# 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

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