Generalized approximately finite $W^*$-algebras
Yosinao Misonou · Tohoku Mathematical Journal · 1955
J. von Neumann has classified factors in some classes, type I, II, III and 1 finite, infinite in his monumental works "Rings of operators".The really interesting is the theory of type II L (type II and finite).As a special one of such factors, F. J. Murray and J. von Neumann [8] investigated approximately finite factors and they have many interesting results.I. Kaplansky [6] generalized this theory to general T7*-algebras.In these theories, the separability condition for the underlying Hubert space is essential.The purpose of this paper is to generalize these theories in non-separable casss, on the basis of the theory of direct products of W*-algebras in the preceding paper[7J.In the first section we shall study some preliminary lemmas.The secondsection will be devoted to the study of factors.A factor will be called to be approximately finite if it is of type Hi and generated by a family of subfactors of type I which mutually commute.Then two approximately finite factors are algebraically ^-isomorphic to each other if and only if the cardinals of families of subfactors mentioned above are identical.Murray and von Neumann's approximately finite factor is considered as a special one.In the final section, we shall generalize the above considerations for factors to general flP-algebras.A TF*-algebra will be called to be approximately finite if it is of type Π t and generated by T7*-subalgebras of type Γ which mutually commute.Especially if these W *-subalgebras have no commutative part in every central decomposition, then it is called to be uniformly approximately finite.Then every approximately finite TF*-algebra can be represented as a direct sum of uniformly approximately finite W *-subalgebras, and a uniformly approximately finite TF*-algebra is a direct product of an approximately finite factor and a commutative ΫF*-algebra.