A Remark on a Preceding Paper by J. Von Neumann

In a preceding paper 'On rings of operators, III" J. von Neumann has developed a theory of normed operators for operators in an arbitrary factor 9R.2 In part the procedure is as follows: First [[A]], the norm of A, is defined for all A e tR of finite rank (Definition 1.3.1) and fundamental sequences of such operators and an equivalence relation for fundamental sequences are defined (Definitions 1.4.1, 1.4.2 and 1.4.3). Then the key theorem (Theorem III) is proved that every fundamental sequence is strongly convergent operatorially, that two fundamental sequences converge strongly to the same limit if and only if they are equivalent, and finally that the norms of a fundamental sequence converge numerically to a limit depending only on the operator limit of the sequence. The proof of Theorem III makes use of two important lemmas: LEMMA 1.4.1. If An, n = 1, 2, * * is a fundamental sequence with [[AJ]] + 0 as n -oo then An converges strongly to 0 as n -* ??. LEMMA 1.4.3. If An, n = 1, 2, * * is a fundamental sequence with An converging strongly to 0 as n -* , then [[An]] -0 as n + oo. The purpose of the present note is to point out how the proofs of these two lemmas can be simplified by using more fully the results of the paper 'On rings of operators, II' by Murray and von Neumann.3 We require the following lemma. LEMMA. If OR is a factor in a finite case and An E OR with An converging strongly to O as n -oo , then [[An]] -*0 as n + oo . PROOF. There is a finite set of elements gi, * * , Mg,, such that [[An]]2 = Tr9 (A*An) = ET-, (A*Angi, gi) = 2t7l (Angi, Angi) = o l| Ang 112 for all n.4 Since A gi -* 0 as n oo for all i, and m is finite, the lemma follows. PROOF OF LEMMA 1.4.3. Suppose the lemma false, if possible. Then for some e > 0 we could assume that [[An]] > e for all n, by choice of a suitable subsequence. For some fixed no , [[An Ar]] < e/2 for all n, r _ no. From the definition of a fundamental sequence, Ano is contained in some closed linear manifold 91 of finite dimensionality relative to 'DR. Let E be the projection on W1.