Ideals, Faces and Compressions
In this chapter we will establish the connection between projections, ideals and faces in C*-algebras and von Neumann algebras. In the first two sections we will explain how projections and ideals are related to each other, and to faces of the normal state space in the von Neumann algebra case, and to faces of the state space in the C*-algebra case. In the next section we will relate projections and ideals to invariant subspaces of the predual of a von Neumann algebra and the dual of a C*-algebra. In the last section we will give an order theoretic characterization of compressions of a von Neumann algebra, i.e., maps of the form a → pap where p is a projection.
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