Baer ring

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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.

1 Definitions
2 Examples
3 Properties
4 References

[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 038705751X
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