Baer ring

Baer ring

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 annihilators 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.

Definitions

*Given a subset "S" of a ring "R", the left annihilator of "S" is the set "{r ε R : rS = 0}".
*An idempotent in a ring is an element "p" with "p"2 = "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.

Examples

*von Neumann algebras are examples of all the different sorts of ring above.

Properties

The projections in a Rickart *-ring form a lattice, which is complete if the ring is a Baer *-ring.

References

*planetmath reference|id=7864|title=Baer ring
*springer|id=R/r080830|title=Regular ring (in the sense of von Neumann)|author=L.A. Skornyakov
*springer|id=R/r081840|title=Rickart ring |author=L.A. Skornyakov
*springer|id=A/a120310|title=AW* algebra|author=J.D.M. Wright
*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


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