Similar Classification Of Holomorphic Curves And Elliott Invariant

The classification or searching invariant is always one of most important project of the research of Mathematics. In the 1970s, G.A. Elliott proved for the AF-algebra A, the pair of it's K₀ group, semigroup and unit (K₀(A), V(A), [1_A])(called Elliott invariant) is the compeltely invariant of the algebra. Elliott invariant plays a a vital role in the classification of math, especially in the field of C* algebra. The main work of this thesis is it's application in the complex geometry and the classification of AI algebra. Thesis is divided into two parts, the first is about the classification of holomorphic curves; the second is about the classification of AI algebras. The two parts have in common is the use of Elliott invariant as the invariant of classification theorem. In 1978, classification of holomorhic curves is first proposed by Mathematicians- M.J.Cowen and R.G.Douglas. They use the knowledge of complex geometry and operator theory to rebuild the Clabi rigid theorem and define a kind of curvaturefunction. And they proved this curvature function can be seen as the unitray invariant of holomorphic curves. The natural question is What is holomorphic curves similar invariant? M.J.Cowen and R.G.Douglas conjecture that thecurvature functionis also the similar invariant of one dimensional holomorphic curve. But people find the conjecture is not reasonable by some examples. In the first part, we give the definition of commutant of holomorphic curve and prove that the K₀ group and ordered K₀ group of the commutant is just the completely similar invariant of one demensionalcurve and most of n-demensional curves. As we all known, from the Swain theorem which is about the vector bundle on paracompact manifold X, the two projectionsin M_∞(C(X)) is equivalent if and only if the two bundles induced by them are the same. It also proves that the isomorphism class of the vector bundle on X are consistent to the equivalent class of projections of M_∞(C(X)). So the relatioship of algebraic K-theory and topologic K-theory is established. But the Swain theorem for holomorpbic complex bundles in complex geometry does not exist. In this paper, the classification theorem reduces the similar equivalence of two holomorphic curves f, g to calculation of the K₀ group of the commutant of f(?)g. In this sense, we build the Swain theorem of complex holomorphic bundles.The second part belongs to the research of classification of C* algebra, An approximateinterval algebra (abbreviated AI algebra) is a separable C*-algebra which is an inductive limit of finite direct sum of matrix algebras over C[0, 1],i.e. (A_n =(?)M_[n,i](C[0,1]))In 1991, George Elliott classified the simple unital approximate interval algebras using an invariant consisting K₀ theory and tracial state data(see [El 2] or [Ste]). In 1995, Kenneth H.Stevens[Ste] 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. In the second part of this paper, our purpose is to 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) will be removed. Let us point out that our proof is completely different from Steven's proof of his theorem. In his proof, Steven introduced a lot of concepts. Most of those concepts heavily depend on the condition-the spectrum is interval [0,1] and don't have higher dimensional analogy.In this paper, we proved a dichotomy condition which can be used to avoid all the technicalities of Steven's paper. Let us point out, this dichotomy condition can be generalized to higher dimension. Once the dichotomy theorem is proved, many technique of the simple case can be used in this new setting. We believe that this new approach will be very helpful for the future classification of AH algebras of higher dimensional spectrum.