It will be shown that the equivalence classes of C*-algebra extensions of C(X) relative to a II^,-factor and Hom(A'(Jf), R) are isomor-phic. This provides a proof for the result of Brown, Douglas _and Fillmore [5] on the isomorphism between the former group and Hom(A/^-Y), R). Let % be a separable infinite dimensional Hubert space, £(%) the algebra of all bounded linear operators on %, %(%) the ideal of compact operators, and &(%) the quotient algebra £(%)/%(%). In [3], [4], [5], ExtX was defined as the set of equivalence classes of C*-algebra extensions, 0^%(%)^&-^C(*)^0 for X a compact metric space and C(X) the algebra of continuous complex-valued functions on X, or equivalently as the unitary equivalence classes of unital *-monomorphism t: C(X)-> &(%). It was shown that Ext X is an abelian group and that it is a generalized homology theory. One of the basic facts about Calkin algebra used in the theory of C*-algebra extensions is the Weyl-von Neumann Theorem, which was generalized to semifinite factors by L. Zsido [10]. P. A. Fillmore has extended part of this theory to semifinite factors [6]. That is, let M be a II«,-factor in % with a dimension function d, %(M) the ideal generated by the finite projections, and &(M) the quotient algebra M/%(M). Extw X was defined as the set of unitary equivalence classes of unital *-monomorphisms t: C(X)-» &(M). An extension t is trivial if t can be factored through M, that is, if there exists a unital *-monomorphism a of C (X) into M such that t = ir ■ a, where it is the natural homomorphism of M onto M/%(M). The sum of t, and r2 is the extension defined as follows: choose isometries Vx and V2 in M such that VXV* + V2Vl = l,and let (T, + t2)(/) = *(Vx)Tx(f)?r(Vt) + *(K2)t2(/MK2*) for all/in C(X). The equivalence class of t, + t2 is independent of choice of isometries in the definition, trivial extensions form the identity element, and

This study analyzes the effect of social support perceived by adolescent in Institutional Care adolescents their well-being. These children have experienced many stressful events and are placed in institutional care with unfinished conflicts. The survey data was collected from 163 adolescents in institutional care facilities. Frequency analysis, reliability analysis, correlation analysis, and multiple regression analysis were applied using the SPSS program. As a result, it was confirmed that the influence of social support perceived by adolescents on adolescent's well-being was positively and significantly affected by all sub-factors of social support, such as parental support, teacher support, and friend support. In particular, as for the influence of social support, it was confirmed that the teacher support was most effective on the adolescent well-being which was followed by parent support and friend support. This means that the higher the sub-factors of social support perceived by adolescents such as teacher support, parental support, and friend support, the higher the adolescent's sense of well-being appears. This study is considered to be basic data for finding ways to promote youth's well-being by understanding the influence of youth's social support on their well-being.

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The purpose of this study is to identify the effects of grit of adolescents at-risk on mindfulness and interpersonal relationship tendency. Therefore, this study focused on the temporal relationships experienced by at-risk adolescents and analyzed the relationship between grit, mindfulness, and interpersonal tendencies of at-risk adolescents who enrolled and trained in psychological counseling centers. The subjects of this study were 170 psychological equivalent students. Frequency analysis, reliability analysis, correlation analysis, and multiple regression analysis were applied using the SPSS program. The results of the analysis are as follows. First, the grit of adolescents at risk was found to have a positive effect on mindfulness. Second, it was confirmed that the grit of adolescents at risk had a positive effect on interpersonal relationship tendency. The purpose of this study is to analyze the effects of grit on mindfulness and interpersonal relationship tendency of adolescents at risk. It is considered to be a basic data for finding ways to improve mindfulness and interpersonal relationship tendency of adolescents at risk.

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