By Isac G.

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**Example text**

Take an arbitrary A E 3,4+. VCe put. r]~ = v ~ , n E N. t. t, the topology p. This completes the proof. We next investigate the continuity of trace functionals with respect to the uniform topology T~. We define the locally convex topology Tc called precompact topology determined by the family of seminorms: p~,~(x)= sup t(x~l~)l, xcM, where ~J~ and ~ range over the precompact subsets of T~[t~]. 11. Suppose 73[t~] is a Fr6chet space. Then every trace functional fT, T E ~1(A4), is continuous with respect to the topology T~.

T. s proof. 6. T h e following s t a t e m e n t s are equivalent: (i) A4 is self-adjoint. (ii) I~I(Ad ) = 1 G ( M ) . P r o o f . 3, (2). (ii) =* (i) Take an a r b i t r a r y ~ E 79"(M). T h e n we have ~ @ ~ E I ~ ( M ) G I ( M ) , which implies ~ E 79. Hence M is self-adjoint. = For any T E 1 G ( A J ) we define two linear functionals on Ad by T f ( X ) = tr T X , f T ( X ) = tr Xt*T, X E M. 7. (1) fT = T f for each T E ~ l ( d ~ ) . A/l)+. (3) Any fT, T E ~l(Ykd), is written as fT = fT1 -- fT2 + i(fTa -- fT4) whereby T j c G](A4)+,j = 1,...

2. Let A4 be the O*-algebra generated by the position and momentum operators Q and P on the Schwartz space S ( ~ ) and put A = ~ ( Q + iP). It can be checked that (AtA - I ) ( A t A - 2I) r P(A4). Hence there exists a positive linear functional f on M such that f((AtA - I ) ( A t A - 21)) < O. 1) Let {~n} be the orthonormal basis in L2(R) consisting of the eigenvectors of the number operator N = AA t. 1) that f is not strongly positive. Many important examples of states in quantum physics are trace functionals, that is, they are of the form f(X) = tr TX with a certain positive trace class operator T.

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