Unbounded Operator Algebras and Representation Theory by K. Schmüdgen
By K. Schmüdgen
*-algebras of unbounded operators in Hilbert house, or extra in general algebraic platforms of unbounded operators, ensue in a average means in unitary illustration thought of Lie teams and within the Wightman formula of quantum box idea. In illustration thought they seem because the photos of the linked representations of the Lie algebras or of the enveloping algebras at the Garding area and in quantum box conception they happen because the vector area of box operators or the *-algebra generated by way of them. the various uncomplicated instruments for the overall idea have been first brought and utilized in those fields. for example, the concept of the susceptible (bounded) commutant which performs a basic position in thegeneraltheory had already seemed in quantum box concept early within the six ties. however, a scientific examine of unbounded operator algebras begun in simple terms first and foremost of the seventies. It used to be initiated via (in alphabetic order) BORCHERS, LASSNER, POWERS, UHLMANN and VASILIEV. J1'rom the very starting, and nonetheless this day, represen tation idea of Lie teams and Lie algebras and quantum box conception were fundamental assets of motivation and likewise of examples. despite the fact that, the overall conception of unbounded operator algebras has additionally had issues of touch with a number of different disciplines. In particu lar, the idea of in the community convex areas, the idea of von Neumann algebras, distri bution idea, unmarried operator idea, the momcnt challenge and its non-commutative generalizations and noncommutative likelihood conception, all have interacted with our subject.
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This set is denoted by 11'(:1), 3e). 2. An O-vector space is an O-family rA such that the operator (Xa + fJb is in rA for arbitrary operators a, b in rA and complex numbers (x, fl. Recall that ab denotes the composition of operators a and b. That is, if a and bare operators on :1) and b:1) ~ :1), then ab is the operator with domain :1) defined by abrp = a(brp), rp E :1). 3. An O-algebra is an O-vector space rA such that b:1)(rA) ~ :1)(rA) and ab ErA for all a, b in rA. 3* 36 2. O-Families and Their Graph Topologies With the addition and scalar multiplication of operators, each O-vector space is a (complex) vector space.
In particular, 1)OO(a) is dense in X. = Proof. There is no loss of generality to assume that X+ ker (a* - i) is finite dimensional. ) Let u be the Cayley transform of a. We extend u to the whole X by defining it to be the zero operator on X+. By a slight abuse of notation, we denote this operator again by u. For n E JNo, let qn+l be the projection of X onto the finite dimensional linear subspace J'n+l := X+ + u*H+ + ... I(a i)n·1I on 1)(an). :D(a") = (I - u)" (I - q,,) X for n E IN. We prove this by induction on n.
Another slight reformulation is the following. ;l is the weakest locally convex topology on 2>(Jl) which makes the embedding of 2>(Jl) into the normed space (2)(Jl), 1I·lIa + 11·11) continuous for each a E Jl. g. SCHAFER , II, § 5). 2. , each a E ,ll is a contimwlts mapping 0/ the locally convex space 2)0« into itself. Proof. )(cA) and a, b E cA. 3. For any O*-algebra cA, 1'+(;])0«) is an O*-algebm on the domain ;])(cA). It is the largest O*-algebra on ;])(cA) whose graph topology coincides with the graph topology 0/ cA.