By Marc A. Rieffel
ISBN-10: 0821825755
ISBN-13: 9780821825754
This paintings describes a basic development of a deformation quantization for any Poisson bracket on a manifold which comes from an motion of $R^d$ on that manifold. those deformation quantizations are strict, within the feel that the deformed fabricated from any features is back a functionality and that there are corresponding involutions and operator norms. the various recommendations concerned are tailored from the idea of pseudo-differential operators. the development is proven to have many favorable houses. a couple of particular examples are defined, starting from easy ones similar to quantum disks, quantum tori, and quantum spheres, to points of quantum teams.
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8 LEMMA. Let {G\} be a net of elements ofBA which converges to G e BA uniformly on compact subsets of V and dominated over all of V by a constant for each semi-norm uniformly in A. Then G\ Xjh converges to G xj h in the same way, for each h G SA; and ( / , G \ Xj K)A converges to (/, G Xj K)A for each f G SA. 4. The second assertion then follows immediately. 9 PROPOSITION. For G e BA and f,h e SA we have (fxjG,h)A = (f,Gxjh)A. 8. 7 (/ xjG,h)A as needed. = lim(/ xj(Gipm),h)A = lim(/, {Gipm) Xjh)A = (f,Gxjh)A, • To conclude this chapter, we derive a useful formula for the deformed product of three functions.
13 it will then be an approximate identity for Cj, and so has the desired property with respect to Aj. But, of course, it is contained in Bj. • We continue by extending a to Aj. 5 that a gives an action of V as automorphisms of A°° for the product Xj, and that this action is strongly continuous for the Frechet topology. 7 it gives an automorphism of Aj. 10 it follows that this action is strongly continuous for the operator norm on Aj. Another way to see all this is to use the embedding of A°° into BA by a i—• a, equivariant for a and r, and the fact that r on BA corresponds to conjugation on SA by translation, which is "unitary" and "strongly continuous".
LeJJ)A + (fJ)A, D For later use there is one more detail which we need to clarify here. Let r denote the action of V on CU(V,A). 8 to this algebra and action to get a C*-norm on the space of smooth vectors, that is, on BA. We will need later the fact that this norm is the same as that on BA defined directly by its action on the A- rigged space SA by Xj. Unwinding the definitions, we see that the former norm is defined as follows. Define F by F(t) = rt(F), so that F can be viewed as a function on V x V.
Deformation Quantization for Actions of rd by Marc A. Rieffel
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