Deformation Quantization for Actions of rd - download pdf or read online

By Marc A. Rieffel

ISBN-10: 0821825755

ISBN-13: 9780821825754

This paintings describes a normal building 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 services is back a functionality and that there are corresponding involutions and operator norms. the various concepts concerned are tailored from the speculation of pseudo-differential operators. the development is proven to have many favorable houses. a few particular examples are defined, starting from simple ones resembling quantum disks, quantum tori, and quantum spheres, to elements of quantum teams.

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Let BA denote the completion of BA for its operator norm. We then form the space S(V,BA), equipped with the structure of a right BA -rigged space. As earlier, let r denote the action of V on we let F be the function in B(V,BA) defined by F(t) = rt(F). We equip B(V,BA) with the product xK, and let it act on S(V, BA) by this product. In particular, F can be viewed in this way as an operator on S(V,BA). The norm of F as an element of (BA)K is then, by definition, the norm of this operator (with respect to the BA -rigged structure).

9 to conclude that Aj is an essential ideal in Cj, so that Cj, and thus Bj, are contained in M(Aj). 17 there is a bounded approximate identity {e\} for the Frechet algebra D°°. 17, we see that it can be arranged to be an approximate identity for all of C°°. 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.

Assume that J is invertible (and skew- symmetric), and let 6 = |det(J~ 1 )|. Let f G SA. Then Lf is a bounded operator on the A-rigged space SA, and in fact iii/ii<« 1 / 2 n/iii- Proof. For g G SA we have (Lfg)(x) = = 6 f(x + Ju)e(u • x)g[u) f(x + u)e(J~xu = 6 f f(u)e(J-Xu • x)g{J^u) • x)g(J-x{u - x)). For fixed u define an operator, Tu, on SA by (Tuh){x) = 61/2e{J~1u • x^J-^u - x)). Then this operator is "unitary" for the A-valued inner product, since for h, k G SA we have (Tuh,Tuk)A = 6 Je(J_1« • ar)fc(J _1 (u - x ) ) ' e ( J - 1 « • a:)A;(

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Deformation Quantization for Actions of rd by Marc A. Rieffel

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