By J.P. Jurzak
Non-commutative integration has its beginning within the classical papers of Murray and von Neumann on earrings of operators, and was once brought as a result of unsolved difficulties in unitary crew representations and the elucidation of assorted elements of quantum-mechanical formalism, including formal calculus in such operator earrings. those papers emphasised the curiosity in 1I -factors and mentioned the impressive habit and 1 algebraic constitution of the set of all unbounded closed operators a. ffiliated to such earrings. The absence of strength instruments in sensible research - normally settled of their definitive shape by means of A. Grothendieck round 1950-195- including the pathological manipulation of algebraic operations on closed operators in Hilbert areas, has restricted ring-theory to the research of algebras of bounded operators with the most aim the tricky query of classifica tion as much as isomorphisms of things. This fabric has authorised a rigorous research of discrete platforms in statistical mechanics yet seems to be much less convincing in different domain names of physics (in the algebraic method of box thought, for example). The impressive function of Hamiltonians, Schrodinger operators and Lie workforce invariant homes in such components of physics disappears within the so known as C*-approach.
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U(Dom(T» c Dom(T) and UTx = TUx for all x in Dom(T). 2. If t E B corresponds to some operator T such that Dom(T) ~ V and T(V) c V (via relation t(x, yl = (Tx, y) for x, y E V), then it follows from (T Bx, y) = (Tx, B*y), with B E M', that T with domain V is affiliated to M (since B(V) c V and B*V c V). 1 shows that such a situation is not always possible, namely, that there exists t in B which cannot correspond to operators T defined on V. 4. Let T be closeable with dense domain Dom(T), and B a bounded operator such that BT c TB.
N -n Id implies ~j ~ ~k ~ Id since N is -n -n -n ~k ~j ~ Aj leads to -2n abelian, hence ~j ~ -n Vkj = nn~1~j (H) = Vj • The same calculation shows that V = np~kvp. £ Thus ~k(Vj) Vj for £ ~£(n V) n ~£(V) c n V k p~k P c p~k k p p~k p of ~~, we get A~V =V for £ = ±1 implies V and, from injectivity ±1, hence, for £ E 1. Each Vp is essentially dense relative to N, as well as V = npvp by [17] . , 2°/. 13. Let M be a von Neumann algebra with commutant M', and An (resp. 1 ~ ~~1 ~ Id (resp. 1 S V~1 SId) and ~~1 E M -n (resp.
A. J~ J~ = BAkC* x AkC* is in C, since BAk x Ak E C . The proof is similar for BAj1 x Aj1 40 Now let T T* E Aid and P n be an increasing sequence of projectors in A'd' with p (H) c Pn n 1 ~ V, ,. It is easily seen that T is the Hilbert ultraweak limit of the sequence Pn T Pn' so that we need to show that TPn x Pn E C. Taking i ~ 0, we see that T PnAi x PnAi is a bounded operator, since (AiP n ) and T are so. '. Let A be an ultraweakly closed subspace of B(V, V) satisfying condition II, with V essentially dense.