On the non-separability of singular representation of operator algebra
Masamichi Takesaki · Kodai Mathematical Journal · 1960
In [3], Feldman and Fell have raised the question whether any separable representation of a TF*-algebra without the direct summand of finite type I is always <τ-weakly continuous or not and they have shown that this is almost affirmative, but the case of type Hi (that is, finite and of type II) remains open.The purpose of the present note is to settle this remainning problem in its positive sense.We have investigated, in [7], the conjugate space of operator algebra and have given an alternative proof of some parts of their above results.We shall use the notation and the result in [7],Bofore going into discussions, the author wishes to express his hearty thanks to Prof. H. Umegaki and Mr. J. Tomiyama for their many valuable suggestions in the presentation of this note.In the proof of our theorem we shall also use the following lemma which played an essential role in [3].LEMMA.Let S be the set of all sequences of integers J= {j\, j 2 ,•} such that l^j n ^ 2 n for each n.Then there exists a subset S 0 of the power of the continuum, such that, for any two distinct sequences J, J r in So, the set of all n for which j n = j n ' is finite.