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Báo cáo toán học: "Pathology in the Calkin algebra "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Bệnh lý trong các đại số Calkin. | J. OPERATOR THEORY 2 1979 159-167 Copyright by ÍNCREST 1979 PATHOLOGY ĨN THE CALKIN ALGEBRA JOEL ANDERSON The Calkin algebra 2 is the quotient of J the bounded linear operator acting on a separable Hilbert space ye by the ideal Jf J of compact operators on ye. Since its appearance in 7 it has been known that the Calkin algebra has many pathological properties. For example it does not contain an infinite increasing sequence with a least upper bound but it does contain an uncountable set of mutually orthogonal projections. However while the known facts indicate that the structure of the Calkin algebra is complicated few concrete results have been obtained. For example a natural problem is to try to classify the masas the maximal abelian selfadjoint subalgebras in the Calkin algebra. The main result along these lines is due to Johnson and Parrott 11 who showed that if srf is a masa in ăă .yey then vfj is a masa in the Calkin algebra. Throughout this paper V shall denote the quotient map of ăă ye onto 2 yey Apparently the only other information we have in this regard is that there is a masa in the Calkin algebra that is not generated by its projections 6 p. 126 and therefore since such a masa cannot lift to a masa in 3d ye the Johnson-Parrott result does not give a complete classification. A masa in the Calkin algebra with special properties is constructed in 5 but it is not known if it lifts to a masa in JfT- The results in this paper show that the Calkin algebra contains other masas with special properties and provide more evidence of the pathological nature of the Calkin algebra. In particular they seem to indicate that the problem of classifying masas is difficult. In section 1 it is shown that assuming the continuum hypothesis there is a masa in the Calkin algebra that is generated by its projections but does not lift to a masa in . This answers a question raised in 5 . Section 2 contains the surprising fact that if f is any state not necessarily pure on the .