A Complete Classification Of Limits Of Splitting Interval Algebras With The Ideal Property

The classification or searching invariant is always one of most important project of the research of Mathematics.In 1991,George Elliott classified the simple unital approximate interval algebras using an invariant consisting K₀ theory and tracial state data(see[1]).Then X.Jiang and H.Su proved that the unital simple imit of finite direct sums of splitting interval algebras can be classified by the same invariant.In 1995,K.H.Stevens[2]proved a generalization of this result by permitting the algebra are unital and have the ideal property(every closed two-side ideal of the algebra is generated by its projections).Furthermore,the algebra is also assumed to be approximately divisible.Recently,Professor Jiang Chunlan and Doctor Ji Kui generalize the Stevens' result to classify all of the AI algebras with the ideal property-that is,both above restrictions(of being unital and approximately divisible) are removed.In this paper,I advance their result and classifies the limits of splitting interval algebras with ideal property by Elliott's invariant, where the splitting interval algebra is a subalgebra of the interval algebra.We deal with the spectrum points in non-Hausdorff topology by discrete topology which is suitable and convinient.And we prove the result about the spectrum of homomorphisms between the splitting interval algebras;that is if there is a point yo∈[0,1]such that SPÑ„_yo contains a fractional point,denoted by 0₁,then 0₁∈SPφ_y,(?)_y∈[0,1](c.f.the theorem 2.4). Thanks to this theorem,we settle many problems at the extreme points.Using the skills just mentioned we proved the existence and uniqueness theorem,then we get the approximately intertwining diagram.By Eiliott's well-known theorem,we get our conclusion.