Let E be a row-finite directed graph, let G be a locally compact abelian group with dual group G = F, let w be a labeling map from E＊ to F, and let （C＊（E）, G,a^w） be the C＊-dynamical system defined by w. Some mappings concerning the AF-embedding construction of C＊ （E） X（aw） G are studied in more detail. Several necessary conditions of AF-embedding and some properties of almost proper labeling map are obtained. Moreover it is proved that if E is constructed by attaching some l-loops to a directed graph T consisting of some rooted directed trees and G is compact, then oJ is k almost proper, that is a sufficient condition for AF-embedding, if and only if ∑j^Kk=1^wγ j ≠fi 1r for any loop γi, γ2 …γk attached to one path in T
Let E be a Hilbert C＊-module,and Y be an orthogonally complemented closed submodule of E.The authors generalize the definitions of Y-complementability and Y-compatibility for general（adjointable） operators from Hilbert space to Hilbert C＊-module,and discuss the relationship between each other.Several equivalent statements about Y-complementability and Y-compatibility,and several representations of Schur complements of Y-complementable operators（especially,of Y-compatible operators and of positive Y-compatible operators） on a Hilbert C＊-module are obtained.In addition,the quotient property for Schur complements of matrices is generalized to the quotient property for Schur complements of Y-complementable operators and Y＊-complementable operators on a Hilbert C＊-module.
