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Let be a fixed separable, infinite dimensional complex Hilbert space, and let () denote the algebra of all (bounded, linear) operators on . The ideal of all compact operators on will be denoted by and the canonical quotient map from () onto the Calkin algebra ()/ will be denoted by π.
Some open problems in the theory of extensions of C*-algebras (cf. ) have recently motivated an increasing interest in the class of all operators in () whose self-commuta tor is compact.
Throughout this paper will denote an infinite dimensional, separable complex Hilbert space, and will denote the unit sphere of (i.e. ). Also will represent the algebra of all bounded linear operators on , and will represent the ideal of all compact operators on . Furthermore will denote the set of all (orthogonal) projections on and will denote the sublattice of consisting of all finite rank projections. In most of the cases (especially when limits are involved) will be regarded as a directed set with the usual order relation inherited from .
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