Baer *-ring

In abstract algebra and functional analysis, 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.

In the literature, left Rickart rings have also been termed left PP-rings. ("Principal implies projective": See definitions below.)

In operator theory, the definitions are strengthened slightly by requiring the ring R to have an involution ${\displaystyle *:R\rightarrow R}$. Since this makes R isomorphic to its opposite ring R^op, the definition of Rickart *-ring is left-right symmetric.

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

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