Baer ring
From Wikipedia, the free encyclopedia
In mathematics, Baer rings, Baer *-rings, Rickart rings, Rickart *-rings, and AW* algebras are various attempts to give an algebraic analogue of von Neumann algebras, using axioms about annhilators of various sets.
Any von Neumann algebra is a Baer *-ring, and much of the theory of projections in von Neumann algebras can be extended to all Baer *-rings, For example, Baer *-rings can be divided into types I, II, and III in the same way as von Neumann algebras.
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[edit] Definitions
- An idempotent in a ring is an element p with p2 = p.
- A projection in a *-ring is an idempotent p that is self adjoint (p*=p).
- A (left) Rickart ring is a ring such that the left annihilator of any element is generated (as a left ideal) by an idempotent element.
- A Rickart *-ring is a *-ring such that left annihilator of any element is generated (as a left ideal) by a projection.
- A (left) Baer ring (named after Reinhold Baer) is a ring such that the left annihilator of any subset is generated (as a left ideal) by an idempotent element.
- A Baer *-ring is a *-ring such that left annihilator of any subset is generated (as a left ideal) by a projection.
- An AW* algebra (introduced by Kaplansky) is a C* algebra that is also a Baer *-ring.
[edit] Examples
- von Neumann algebras are examples of all the different sorts of ring above.
[edit] Properties
The projections in a Rickart *-ring form a lattice, which is complete if the ring is a Baer *-ring.
[edit] References
- Baer ring on PlanetMath
- L.A. Skornyakov, "Regular ring (in the sense of von Neumann)" SpringerLink Encyclopaedia of Mathematics (2001)
- L.A. Skornyakov, "Rickart ring" SpringerLink Encyclopaedia of Mathematics (2001)
- J.D.M. Wright, "AW* algebra" SpringerLink Encyclopaedia of Mathematics (2001)
- Sterling K Berberian Baer *-rings ISBN 0-387-05751-X
- I. Kaplansky, Rings of Operators, W. A. Benjamin, Inc., New York, 1968.
- C.E. Rickart, Banach algebras with an adjoint operation Ann. of Math. , 47 (1946) pp. 528–550
