By Gert Kjaergard Pedersen

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**Example text**

Another inequal- ity, (X ± y)*(x ± y) :::: 2(x*x + y*y), shows that n is additive. Therefore, n is a left ideal. M follows from (i). The polarization identity, 3 =L i k (x + iky)*(x k=O implies that m is spanned linearly by p. M+, Xl, ... , Xn, Yl, ... , Yn E n. k=l Since a = a*, the above (3) implies that a =~ t {(Xk + yd*(Xk + Yk) - b= 2 L(Xk + Yk)*(Xk + Yk) (Xk - Yk)*(Xk - Yk)}. k=l So, if we set I n E p, k=l then we have 0 :::: a :::: b, thus a E p. D. vn 42 Weights We apply this lemma to Pq> with ((J a weight on oM.

PROOF: (i) Let {~d be a net in 2l such that {~d converges to ~ in :ott and {7fl(~i)} converges to x E ~l (2l) a -strongly*. Let ~ = lim ~i . For any 17 E 2l', we have = lim7fr(17)~i = lim7fl(~i)17 = X17; 7fr(17)~tt = lim7fr~f = lim1rl(~j)*17 = x*17, 7fr(rO~ so that ~ is left bounded and x = 7fl (~). (ii) This follows from the above same arguments. D. E £(SJ). 25. O f(XiX:> I = f(xx*) in the strong operator topology. Let g(t) = f(t 2 ), t E R, and hi and h be the self-adjoint operators on SJ EB SJ given by the matrices: PROOF: hi = (0 Xi x*) o .

L" = ~ n ~~. l). 15'. l") = nl n ni. l(iv) = ... l'" = . . 16. l". l1 are isometrically *-isomorphic. l under consideration is full. 10'. l' by (21') We then conclude: (i) ao and bo are both preclosed, ao C be; and bo c ae; ; § 1 Left Hilbert Algebras and Right Hilbert Algebras (ii) If nl(~) with 11 o o = a * and ne(~~) = b * , then ne(n and ne(~~) are both affiliated ~e (2t) . We leave the proof to the reader. We now come to the first result which links ~e(2t) and ~e(2t)'. We maintain the previous notations and assumptions, in particular we assume that 2t is a full left Hilbert algebra.