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Description of  Ranges of Bimodule Projections and Conditional Expectations

1443846120 pdf The algebraic theory of corner subrings introduced by Lam (as an abstraction of the properties of Peirce corners eRe of a ring R associated with an idempotent e 2 R) are investigated here in the context of Banach and C_-algebras. We propose a general algebraic approach which includes the notion of ranges of (completely) contractive conditional expectations on C_-algebras and on ternary rings of operators and we investigate when topological properties are consequences of the algebraic assumptions. For commutative C_-algebras we show that dense corners cannot be proper and that self-adjoint corners must be closed and always have closed complements (and may also have non-closed complements). For C_-algebras we show that Peirce corners and some more general corners are similar to self-adjoint corners. We show uniqueness of complements for certain classes of corners in general C_-algebras, and establish that a primitive C_-algebra must be prime if it has a prime Peirce corner. Further we consider corners in ternary rings of operators (TROs) and characterise corners of Hilbertian TROs as closed subspaces. Additional results concern a de_nition we introduce for AW_-TROs which generalize AW_-algebras in a similar way as W_-TROs generalize W_-algebras. In particular, we show that the decomposition of AW_-algebras into type I, II, and III can be used to give a similar decomposition for AW_-TROs. Read more