Ranges of Bimodule Projections and Conditional Expectations
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| Title: | Ranges of Bimodule Projections and Conditional Expectations |
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| Description: | 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 in R) is 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. |
| Authors: | Robert Pluta, Author |
| Resource Type: | eBook. |
| Subjects: | Rings (Algebra), Algebra |
| Categories: | MATHEMATICS / Algebra / Intermediate |
| Database: | eBook Collection (EBSCOhost) |
| Abstract: | 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 in R) is 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. |
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| ISBN: | 9781443846127 9781443867863 |
