The Unitary Equivalence of Operators and K_0-Group of Commutant Algebras

We show that two irreducible operators on H are unitari1y equivalentif and only if W*(A B)’≌M2(C), and give an answer to the open question posedby J. B. Conway (Subnormal Operators, πPitman, Advanced Publishing Program,Boston, London, Melbourne, 1981) for irreducible operator. We also show that ifT, T1 and T2 are irreducible operators with T T1≌T T2, then T1≌T2. Finally,weshow that K0 (A(D))≌Z, giving a new result on the K0-group of Banach algebras.

On K_(0)-groups of operator algebras on Banach spaces

This paper merges some classifications of G-M-type Banach spaces simplifically, discusses the condition of K0(B(X)) = 0 for operator algebra B(X) on a Banach space X, and obtains a result to improve Laustsen's sufficient condition, gives an example to show that X ≈ X2 is not a sufficient condition of K0(B(X)) = 0.

K_0-group and approximate similarity invariants of strongly irreducible operators

Let H olenote a complex separable Hilbert space and L(H) denote the collection of bounded linear operators on H. An operator T ∈ L(H) is said to be strongly irreducible if T does not commute with any nontrivial idempotent. Herrero and Jiang showed that the norm-closure of the class of all strongly irreducible operators is the class of all operators with connected spectrum. This result can be considered as an "approximate inverse" of the Riesz decomposition theorem. In the paper, we give a more precise characterization of approximate invariants of strongly irreducible operators. The main result is: For any T ∈ L(H) with connected spectrum and ε > 0, there exists a strongly irreducible operator A, such that A - T < ε, V (A (A)) =～ N, K0(A (A)) ～= Z, and A (A)/rad A (A) is commutative, where A (A) denotes the commutant of A and rad A (A) denotes the Jacobson radical of A (A). The research is inspired by the recent similarity classification technique of Cowen-Douglas operators of Jiang Chunlan.

K_0-group and similarity of operator weighted shifts

By using simultaneous triangularization technique, similarity for operator weighted shifts with finite multiplicity is characterized in terms of K0-group of commutant algebra. The result supports the conjecture posed by Cao et al. in 1999. Moreover, we discuss the relations between similarity and quasi-similarity for operator weighted shifts.

Unitary Equivalence,Similarity and Calculation of K_0-Group

In this paper, we are concerned with the classification of operators on complex separable Hilbert spaces, in the unitary equivalence sense and the similarity sense, respectively. We show that two strongly irreducible operators A and B are unitary equivalent if and only if W＊（A＋B）′≈M2（C）, and two operators A and B in B1（Ω） are similar if and only if A′（AGB）/J≈M2（C）. Moreover, we obtain V（H^∞（Ω,μ）≈N and Ko（H^∞（Ω,μ）≈Z by the technique of complex geometry, where Ω is a bounded connected open set in C, and μ is a completely non-reducing measure on Ω.

Category X(R) and A-groups

New groups: X-groups are constructed. These groups are easy to calculateand they reflect more general properties of rings than K0-groups. As their applications, some characteristics of group rings with torsion-free K0-groups are obtained.

Classification Theory for Inductive Limit C^＊-algebras

This paper is a survey on the recent work of the authors and their col-laborators on the Classification of Inductive Limit C*-algebras. Some examples are presented to explain several important ideas.

Similarity invariants of operators on a class of Gowers-Maurey spaces

This paper studies the similarity invariants of operators on a class of Gowers-Maurey spaces, ∑dc spaces, where an infinite dimensional Banach space X is called a ∑dc space if for every bounded linear operator on X the spectrum is disconnected unless it is a singleton. It shows that two strongly irreducible operators T1 and T2 on a ∑dc space are similar if and only if theK0-group of the commutant algebra of the direct sum T1 GT2 is isomorphic to the group of integers Z. On a ∑dc space X, it uses the semigroups of the commutant algebras of operators to give a condition that an operator is similar to some operator in （∑SI）（X）, it further gives a necessary and sufficient condition that two operators in （∑SI）（X） are similar by using the ordered K0-groups. It also proves that every operator in （∑SI）（X） has a unique （SI） decomposition up to similarity on a ∑dc space X, where （∑SI）（X） denotes the class of operators which can be written as a direct sum of finitely many strongly irreducible operators.

The commutant and similarity invariant of analytic Toeplitz operators on Bergman space

The famous von Neumann-Wold Theorem tells us that each analytic Toeplitz operator with n + 1-Blaschke factors is unitary to n + 1 copies of the unilateral shift on the Hardy space. It is obvious that the von Neumann-Wold Theorem does not hold in the Bergman space. In this paper, using the basis constructed by Michael and Zhu on the Bergman space we prove that each analytic Toeplitz operator M B(z) is similar to n + 1 copies of the Bergman shift if and only if B(z) is an n + 1-Blaschke product. From the above theorem, we characterize the similarity invariant of some analytic Toeplitz operators by using K 0-group term.

Some Properties of Completely Arithmetical Rings

In this paper, we introduce the concept of completely arithmetical rings and investigate their properties. In particular, we prove that if R is a completely arithmetical ring with J（R） =0, then Ko（R） ≌Z^n for some positive integer n. We also show that such a ring is precisely a ring in which every proper ideal can be written uniquely as a product of finitely many distinct completely strongly irreducible ideals.