The Index Semigroup and K-Groups of Graph-Groupoid C^{*}-Algebras

In this paper, we consider the relation between index theory and $K$ -theory induced by directed graphs. In particular, we study index-morphism on finite trees, and classify the set of finite trees in terms of our index-morphism. Such a morphism generate certain semigroup, called the index semigroup. From the index semigroup, we find a ple, interesting connection between semigroup-elements and $K$ -group computations of groupoid $C^{*}$ -algebras generated by graphs. In conclusion, we show that the pure combinatorial data of graphs completely characterize and classify the elements of the index semigroup (or equivalently, graph-index on finite trees), Watatani’s Jones index on groupoid $C^{*}$ -algebras generated by finite trees, and $K$ -group computations of certain $C^{*}$ -algebras.     

C^＊-代数中的等价

在算子理论和算子代数的研究中，等价是一个很重要的工具，它对K-理论的研究和理解起着很重要的作用．寻找等价关系是研究不变量的一种方法，为了使大家更好的认识和学习C^＊-代数，我们在本文研究了算子代数中经常用到的代数等价、酉等价、同伦等几种等价之间的关系．     

Extremal matrix states on tensor product of C^{*}-algebras

A classic result of Namioka and Phelps on the extreme points of the state space on the tensor product of order unit spaces is generalized to the setting of matrix convexity. We show that the matrix extreme points of the matrix state space on the tensor product of two unital \(C^*\) -algebras, at least one of them is of type \(\mathrm {I}\) , are exactly the isometric orbit of the restrictions, on the exponentially unitary orbit of the \(C^*\) -algebraic tensor product, of the tensor products of the matrix extreme points of matrix state spaces of the two \(C^*\) -algebras.     

On the Universal C*-Algebra Generated by a Partial Isometry

A universal C*-algebra is constructed which is generated by a partial isometry. Using grading on this algebra we construct an analog of Cuntz algebras which gives a homotopical interpretation of KK-groups. It is proved that this algebra is homotopy equivalent up to stabilization by 2×2 matrices to M ₂(C). Therefore those algebras are KK-isomorphic.     

Quotient Algebras of Toeplitz-Composition C^{*}-Algebras for Finite Blaschke Products

Let $R$ be a finite Blaschke product. We study the $C^*$ -algebra $\mathcal TC _R$ generated by both the composition operator $C_R$ and the Toeplitz operator $T_z$ on the Hardy space. We show that the simplicity of the quotient algebra $\mathcal OC _R$ by the ideal of the compact operators can be characterized by the dynamics near the Denjoy–Wolff point of $R$ if the degree of $R$ is at least two. Moreover we prove that the degree of finite Blaschke products is a complete isomorphism invariant for the class of $\mathcal OC _R$ such that $R$ is a finite Blaschke product of degree at least two and the Julia set of $R$ is the unit circle, using the Kirchberg–Phillips classification theorem.     

Extensions of Semigroups and Morphisms of Semigroup $ C^{*} $ -Algebras

Siberian Mathematical Journal - The paper is devoted to the normal extensions of discrete semigroups and $ * $ -homomorphisms of semigroup $ C^{*} $ -algebras. We study the normal extensions of...     

Crossed Product of a C *-Algebra by a Semigroup of Endomorphisms Generated by Partial Isometries

The paper presents a construction of the crossed product of a C^*- algebra by a semigroup of endomorphisms generated by partial isometries. This work was in part supported by Polish Ministry of Science and High Education grant number N N201 382634.     

On Multiplication of Distributions Generated by the Pommiez Operator

Mathematical Notes -     

Nilpotency of the Alternator Ideal of a Finitely Generated Binary (-1,1)-Algebra

We prove nilpotency of the alternator ideal of a finitely generated binary (-1,1)-algebra. An algebra is a binary (-1,1)-algebra if its every 2-generated subalgebra is an algebra of type (-1,1). While proving the main theorem we obtain various consequences: a prime finitely generated binary (-1,1)-algebra is alternative; the Mikheev radical of an arbitrary binary (-1,1)-algebra coincides with the locally nilpotent radical; a simple binary (-1,1)-algebra is alternative; the radical of a free finitely generated binary (-1,1)-algebra is solvable. Moreover, from the main result we derive nilpotency of the radical of a finitely generated binary (-1,1)-algebra with an essential identity.     

On the Structure of the C *-Algebra Generated by Toeplitz Operators with Piece-wise Continuous Symbols

We study the C ^*-algebra generated by Toeplitz operators with piece-wise continuous symbols acting on the Bergman space on the unit disk in . We describe explicitly each operator from this algebra and characterize Toeplitz operators which belong to the algebra. To the memory of G. S. Litvinchuk     

On the Algebra Generated by the Bergman Projection and a Shift Operator I

Let be a domain with smooth boundary and let α be a C ²- diffeomorphism on satisfying the Carleman condition .We denote by the C*-algebra generated by the Bergman projection of G, all multiplication operators aI and the operator where is the Jacobian of α. A symbol algebra of is determined and Fredholm conditions are given. We prove that the C*-algebra generated by the Bergman projection of the upper half-plane and the operator is isomorphic and isometric to . Submitted: February 11, 2001?Revised: January 27, 2002     

The Structure of Fractional Spaces Generated by a Two-dimensional Difference Operator in a Half Plane

Ukrainian Mathematical Journal - We consider a difference-operator approximation $$ {A}_h^x $$ of the differential operator $$...     

The Minimal Process Generated by a Differential Operator of the Second Order in R^n

For a differential operator $\Omega u=\sum\limits_i,j=1^n \frac{\partial}{\partial x_i}(a_ij(x)\frac{\partial u}{\partial x_j})+\sum\limits_{i=1}^n b_i(x)\frac{\partial u}{\partial x_i}+c(x)u$ with unbounded coefficients in R^n, a standard continuous paths process with infinitesimal operator \Omega has been constructed in this paper, and the invariance of such process under a transformation group of phase space has been discussed.     

平削算子生成的B值鞅空间及其原子分解

对于一系列由平削算子生成的Banach空间值鞅空间建立了原子分解定理 ,并以此为工具讨论了它们之间的相互嵌入关系 ,其结果与Banach空间的凸性和光滑性有密切联系 .