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By Boris Fedosov

Within the monograph a brand new method of deformation quantization on a symplectic manifold is constructed. This procedure offers upward thrust to a tremendous invariant, the so-called Weyl curvature, that is a proper deformation of the symplectic shape. The isomophy sessions of the deformed algebras are categorized via the cohomology periods of the coefficients of the Weyl curvature. those algebras have many universal positive factors with the algebra of whole symbols of pseudodifferential operators other than that during normal there are not any corresponding operator algebras. however, the constructed calculus permits to outline the thought of an elliptic aspect and its index in addition to to end up an index theorem just like that of Atiyah-Singer for elliptic operators. The corresponding index formulation includes the Weyl curvature and the standard elements getting into the Atiyah-Singer formulation. purposes of the index theorem are hooked up with the so-called asymptotic operator illustration of the deformed algebra (the operator quantization), the formal deformation parameter h could be changed by means of a numerical one ranging over a few admissible set of the unit period having zero as its restrict element. the truth that the index of any elliptic operator is an integer ends up in useful quantization stipulations: the index of any elliptic point may be asymptotically integer-valued as h has a tendency to zero over the admissible set. For a compact manifold an immediate development of the asymptotic operator illustration indicates that those stipulations also are adequate. ultimately, a discount theorem for deformation quantization is proved generalizing the classical Marsden-Weinstein theorem. thus the index theorem offers the Bohr-Sommerfeld quantization rule and the multiplicities of eigenvalues.

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This implies there are E > 0 and N 2 1 such that n,, - W, = A. Thus We now define a new sequence of open symmetric neighborhoods of A by vo = wo, vn = Wl+(*-l)N, Obviously, n 2 1. V, = A. 11 we must prove that V,+l o Vn+l o V,+l for n 2 0. c V, It is clear that V1 o V1 o V1 c VOand by the choice of E and N , V2 o Vz o VZ C V1. Let k 2 2 and take (z,y) E vk+1Ovk+1Ovk+1. Then we prove that (z,y)E v k . Let z , w E X be points such that (z,w), (w,z),(z,y) E Vk+1. If ( p , q ) is any of these three points, then we have from which d(fi(z),fi(y)) < e, lil < 1 + (k - l)N.

Here d(A,B) is defined by d ( A , B )= inf{d(a,b) : a E A,b E B}. To show (4) let 1 > 0. For 1 5 i 5 m there is 0 < ~i < X such that if d(f(z),f(z)) < ~i (f(z), f(z) E f(cl(Ui))) then d(z,z ) < 7. This follows from the fact that f:cl(Ui) --f f(cl(Ui)) is injective. Put E = min{si : 1 5 i 5 m } . If diam(D) < E , then we have diam(Di) < 1 for 1 5 i 5 k. e. f is k-to-one. - . (n;=, u,"==, In the remainder of this section we describe well known theorems that will be used in the sequel. Let X be a topological space.

If this is false, for any n > 0 there exists a subset A, such that B for all B E a. diam(A,) = sup{d(a,b) : a , b E A,} 5 l / n and A, Take x , E A,, for n 2 1. Since X is compact, there is a subsequence {x,,} such that x,, 4 x as i -+ 00. Then x E B for some B E a. Since X \ B is compact, we have a = d ( x , X \ B ) = inf{d(x,b) : b E X \ B} > 0. Take sufficiently large ni satisfying ni > 2 / a and d(x,, ,x ) < a/2. Then, for y E Ani we have d ( y , ~ ) I d ( ~ i x n i ) + d ( z n i , l~/ n ) Ii + a / 2 < a < and thus y E B.

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