Study On The K-groups Of Operator Algebras B(X)

Master's academic article " Study on the K-groups of operator algebras B(X) " is the organic combinative result of the study on the theory of Banach space and the operator theory of functional analysis.In §1,through leading into the Ka operator and the second Kato spectrum σ_k^'(T),we prove that the set σ_k^'(T) ,contained in σ(T),is a compact subset of C. The next two chapters discuss K-groups of operator algebras B(X) on Banach space X combining the series of G-M results of the structural theory on Banach space.In §2, according to the special operator form on HD_n space and QD_n space,we mainly calculate the K-groups of operator algebras B(X) on the two types of spaces under certain conditions. The section §3 is the main work of this paper. We mainly discuss the K₀-groups of operator algebras B(X), obtain a necessary and sufficient condition of K₀(B(X)) = 0. And we also get a corollary negating Gowers's guess that X ≈ X² decides K₀(B(X)) = 0. The section §3 also gives a sufficient condition of K₀(B(X)) = Z₂.