TAILIEUCHUNG - Báo cáo toán học: "Operators commuting with Toeplitz and Hankel operators modulo the compact operators "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Các nhà khai thác đi lại với các nhà khai thác Toeplitz và Hankel modulo các nhà khai thác nhỏ gọn. | J. OPERATOR THEORY 6 1981 55-74 Copyright by INCREST 1981 POSITIVE MATRICES AND DIMENSION GROUPS AFFILIATED TO C -ALGEBRAS AND TOPOLOGICAL MARKOV CHAINS DAVID HANDELMAN 1. INTRODUCTION In the very recent past the study of c -algebras by means of partially ordered abelian groups has been carried out. For AF-C -algebras dimension groups arise these are direct limits as ordered groups of free finitely generated abelian groups equipped with the pointwise ordering. The maps between the free groups are determined by rectangular matrices with nonnegative integer entries. Given a dimension group it is of interest to know how to construct it explicitly with the rectangular matrices given algorithmically especially since dimension groups can be characterized abstractly . A convincing application of this is the recent embedding result of Pimsner and Voiculescu 13 which asserts that the irrational rotation algebra with angle 2mt Ax may be embedded in an AF-algebra c whose dimension group is Gx z aZ ordered as a subgroup of the reals. This depends on an explicit algorithm for obtaining Gx as an order limit of free groups of rank two due to Effros and Shen 5 This embedding has greatly increased the knowledge of the structure of Ax see 14 for a discussion . A major result of this article is to characterize those totally ordered and somewhat more generally simple free abelian groups arising as the limit group with the same map repeated over and over. These admit an elementary algebraic characterization. The ordered groups that occur in an invariant due to Krieger 11 12 for irreducible subshifts of finite type in the theory of topological Markov chains are precisely these limit groups. Up to order-isomorphism these limits of stationary systems are classified by a triple A r A - R where A is an integral order in a number field cr represents an equivalence class of real embeddings and represents an ideal class. If A is a matrix with strictly positive entries then one of its .

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