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


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать курсовую

Look at other dictionaries:

  • BAER, MAX — (Maximilian Adelbert; 1909–1959), U.S. prize fighter, world heavyweight champion 1934–35, member of the World Boxing Hall of Fame and the International Boxing Hall of Fame. Born in Omaha, Nebraska, Baer dropped out of school in the eighth grade… …   Encyclopedia of Judaism

  • Ring Magazine upsets of the year — Ring Magazine was established in 1922. Since 1980, it has named a fight that has resulted in an outcome that was highly contrary to general expectations as upset of the year, based on the magazine s writers criteria. Here is a list of Ring… …   Wikipedia

  • Ring Magazine round of the year — Ring Magazine was established in 1922. Since 1945, it has named a particular round of a fight as round of the year, based on the magazine s writers criteria. Here is a list of Ring Magazine s rounds of the year:1920s*1922 Leonard Tendler, Round 8 …   Wikipedia

  • Baer, Max — ▪ American athlete in full  Maximilian Adelbert Baer   born February 11, 1909, Omaha, Nebraska, U.S. died November 21, 1959, Hollywood, California  American boxer who won the world heavyweight championship by knocking out Primo Carnera (Carnera,… …   Universalium

  • Ring Magazine fights of the year — Every year since 1945, The Ring magazine has named a fight as fight of the year, based on the magazine s writers criteria. Here is a list of fights of the year as selected by The Ring :Fights of the year by decade1920s*1922 Harry Greb W 15 Gene… …   Wikipedia

  • Baer, Max — (1909 1959)    Born Maximilian Adelbart Baer Cussen to German immigrant parents in Omaha, Nebraska, boxer Max Baer developed his strength and physique working on cattle ranches in Colorado. This power gave him a devastating right hand punch and a …   Historical Dictionary of the Roosevelt–Truman Era

  • Baer, Max (Maximilian Adelbert) — (1909–59)    World heavyweight champion. Born in Omaha, Nebraska, Baer claimed Jewish paternity and appeared in the ring with the Star of David on his trunks. He was world heavyweight champion in 1934, and later became an actor …   Who’s Who in Jewish History after the period of the Old Testament

  • Reinhold Baer — (July 22 1902 – October 22 1979) was a German mathematician, known for his work in algebra. He introduced injective modules in 1940. He is the eponym of Baer rings.Baer studied mechanical engineering for a year at the University of Hanover. He… …   Wikipedia

  • Max Baer (boxer) — This article is about the boxer and actor. For an article about his actor son, see Max Baer, Jr. Max Baer Statistics Real name Maximilian Adelbert Baer Nickname(s) …   Wikipedia

  • Max Baer — Fiche d’identité Nom complet Maximillien Baer Nationalité …   Wikipédia en Français

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”