By von Neumann J.

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I r J V. u V 1 ' ' r í (u,v6N) JJ 1 *' s We set P°(u,v) = uv. For r = 1, we obtain the Poisson operator P. Given a formal function f(z) with constant coefficients such r r that f(o) = 1, substitute Ń in the expansion of f (vz) ; we to ć obtain a bilinear map (u,v)ŁN x N ->u ±^ í = f(vP)(u,v) 6 E(N;v). We wish to choose f so that we define thus a star-product. The answer is given by the following : Proposition - If (W3h^Y) is a flat symplectic manifold, there is a unique formal function of the Poisson bracket that generates a star-product : it is the exponential function.

5 ^ (,0-8) wil show tha the altern (I) 3-cohain a Chevalley 3-cocycle. It follows from (10-8) that we have : l (3) ~ ^ *2q+2 2q+2 L let, according to (10-4) : (3 (10-9) T_ ^ 2q+2 = (2/3) D q It follows from the proposition of b, for the order 2q, that (3) is a Chevalley 3-cocycle and it is the same for T 0 „. Such a co2q+2 cycle is the image of a closed 3-form of W by ě ^ and it is exact if the 3-form is exact. 58 ANDRΙ LICHNEROWICZ (3) d) We search for a modification of C n such that T 0 0 i va2q+l 2q+2 + , we er c o ro rn e s nishs.

Consider also another bilinear map Í x Í responding to the formal series : Γ f (3-6) u *^ í = uv + Σ r=l ν C (u,v) E(N;v) cor­ 36 ANDRE LICHNEROWICZ where the are differential 2-cochains again. Suppose that (3-5) is such that we have formally the identity (3-7) Ô (u *' í) V = V T u * Ô í ν í í By means of universal formulas, we can prove : Proposition - The deformation ( 3 - 1 ) of ( N , . ) being given3 each format series ( 3 - 5 ) where the Tq are necessarily differential ope­ rators 3 generates a unique bilinear map ( 3 - 6 ) satisfying ( 3 - 7 ) ; this map is a new deformation which is sayed to be equivalent to (3-1).

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Collected works. Vol.4 Continuous geometry and other topics by von Neumann J.

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