By J.P. Jurzak
Non-commutative integration has its beginning within the classical papers of Murray and von Neumann on jewelry of operators, and was once brought due to unsolved difficulties in unitary crew representations and the elucidation of varied facets of quantum-mechanical formalism, including formal calculus in such operator jewelry. those papers emphasised the curiosity in 1I -factors and mentioned the striking habit and 1 algebraic constitution of the set of all unbounded closed operators a. ffiliated to such earrings. The absence of energy instruments in practical research - normally settled of their definitive shape through 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 target the tricky query of classifica tion as much as isomorphisms of things. This fabric has authorised a rigorous examine of discrete structures in statistical mechanics yet seems to be much less convincing in different domain names of physics (in the algebraic method of box concept, for example). The impressive position of Hamiltonians, Schrodinger operators and Lie crew invariant houses in such components of physics disappears within the so referred to as C*-approach.
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Additional info for Unbounded Non-Commutative Integration
B2 ) be the abelian *-algebra generated by all (Al)~ (resp. (A2)~)' i ENE = ±', with natural domain V 1 (resp. V2 ). Since the description of V, (resp. V 2 ) depends only on B, (resp. B2 ), one may start directly from Bl' B2 in order to define this tensor product. 2. , V = Vl or V2 ) is of type V = nn~OA-n~H) iff V coincide topologically with the Frechet space Coo(e~tA) of differentiable vectors for the unitary one-parameter itA 00 itAl 00 itA2 group t ......... e . Thus, V l = C (e ) and V 2 = C (e ) 00 itAl itA2 leads to V l ® V 2 = C (e ® e ), since Al ® 1 + 1 ® A2 is the infinitesimal generator of the unitary group itAl itA2 t .........
O HA . 3. , equicontinuous) disked subset B of V'. 3, thus proving our proposition. 5. Let A be a *-algebra satisfying condition 10 , with natural domain V. Let A € A, A ~ Id such that all powers An, n € N are essentially self-adjoint on V, and VA = nn~O Dom(An ) the (Fr~chet) space endowed with semi-norms x € VA ~ IIAnxll , for n € N. Then VA => V and the natural injection from V into VA is continuous for respective topologies, and has a dense image. Proof. Since An € A for n ~ 0, formula V C Fr~chet VA is straight- forward, as well as continuity of the canonical injection.
Conversely, let us assume the existence of an involutive algebra N c L(VJ n M, strongly dense in M. -1 One can suppose that An E N for every n, since the algebra generated by N and the A~1 has the same properties. The *-algebra C generated by N and operators An' with n € clearly dominated by operators An' hence admits IN is V as natural 13 domain. Of course, N c Cid eM, since C c 8 n L(V). It is immediately seen that C satisfy condition I, thus the a-weak closure of C, relative to V 0 V, is the space Un~O(Cid)" An x An = 8.