Ranges of Bimodule Projections and Conditional Expectations

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.
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