On normal $AW^{\ast}$-algebras
Kazuyuki Saitô · Tohoku Mathematical Journal · 1981
An AI7*-algebra M is a C*-algebra with the following two properties: (a) in the set of projections M p in M, every orthogonal collection has a least upper bound, (b) every maximal abelian *-subalgebra is generated by its projections [2].Kaplansky ([2], [3], [4] (see also [1])) showed that much of the "nonspatial theory" of von Neumann algebras can be extended to AW*algebras.Above all, he showed that M p is a complete lattice.One of the difficulties in treating APT*-algebras is that, because of the lack of the strong topology as in von Neumann algebras, there is no guarantees for the fact that whenever {f β } is an increasing net of projections with the supremum / in M p , then / is the supremum of {f β } in the partially ordered space M h of the hermitian part of M.An AW*-algebra M is said to be normal if, for every increasing net {e a } of projections in M with the supremum e in M p , e is the supremum of {e a } in M h (that is, if aeM h such that a ^ e a for all α, then a ^ e) [8].It is known that every monotone complete C*-algebra (a von Neumann algebra, a type 1 A17*-algebra) is normal.In [8], Wright proved the following interesting result, by using the regular ring, to the effect that every finite ,4.17*-algebra is normal (a similar result was also proved by Hamana by using the regular monotone completion of .A ΫP*-algebras [9]).We say that an increasing net {e a } of projections with the supremum e in M p in a C*-algebra M is well-behaved if, whenever x (in M h ) satisfies e a xe a ^ 0 for all α, then exe ^ 0.In this paper, by using the above concept, we shall show the following theorem which is a nominally more general result on the one hand and is a simple alternative proof of the theorem of Wright and Hamana on the other (see Corollary).THEOREM.Let M be an AW*-algebra.Then M is normal if and only if every increasing net of projections in M is well-behaved.