Classification of Real Von Neumann Algebras (I)

Abstract

Let M be a real Von Neumann (VN simply) algebra on a real Hilbert space H, and p, q be two projections of M. Similarly to the complex case ([1]), p ~ q if there exists v ∈ M such that p = v*v and$q = vv * ;p\;\underline \prec \;q$ if there exists a projection q ₁ ∈ M such that p ~ q ₁ and q ₁ ≤ q. Then we have the definitions of finiteness, infiniteness and others for real VN algebras and their projections. And we have the unique decomposition:

M = M₁ ⊕M₂⊕M₃,

where M₁,M₂,M₃ are finite, semi-finite and properly infinite, purel infinite real VN algebras respectively.