COMPARISONS OF EQUIVALENCE RELATIONS ON OPEN PROJECTIONS

Abstract. 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. Zsido ́ 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 C0(Ω)⊗K(ℓ2) exactly when the canonical homomorphism from Cu(C0(Ω)) into Lsc(Ω; N̄0) is bijective. Here, Cu(C0(Ω)) is the stabilized Cuntz semigroup, and Lsc(Ω; N̄0) is the semigroup of lower semicontinuous functions from Ω into N0: = {0, 1, , 2,...,∞}. 1