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Hermitian star products are completely positive deformations. (English) Zbl 1081.53078
Let \(M\) be a smooth manifold and let \(C^{\infty}(M)\) be the algebra of complex-valued smooth functions on \(M\). A positive linear functional on \(C^{\infty}(M)\) is a complex linear functional \(\omega_0 : C^{\infty}(M)\rightarrow {\mathbb C}\) such that \(\omega _0({\bar f}.f)\geq 0\) for all \(f\in C^{\infty}(M)\). Assume that \(M\) is a Poisson manifold and \(\ast\) is a star-product on \(M\) satisfying \(\overline {f\ast g}={\bar g}\ast {\bar f}\) for \(f,g\in C^{\infty}(M)[[\lambda]]\). A \({\mathbb C}[[ \lambda]]\)-linear functional \(\omega_0 : C^{\infty}(M)[[\lambda]]\rightarrow {\mathbb C}[[ \lambda]]\) is called positive if, for each \(f\), \(\omega _0({\bar f}\ast f)\in {\mathbb R}[[\lambda]]\) is positive, i.e., the first non-zero coefficient of \(\omega _0({\bar f}\ast f)\) is positive.
The main result of the present paper asserts that any positive linear functional \(\omega _0\) on \(C^{\infty}(M)\) can be deformed into a positive \({\mathbb C}[[ \lambda]]\)-linear functional \(\omega=\omega_0+\sum_{k\geq 1}{\lambda}^k\omega_k\) on \(C^{\infty}(M)[[\lambda]]\). The proof of this result follows the same steps as the one for symplectic star-products [see H. Bursztyn and S. Waldmann, “On positive deformations of *-algebras”, Math. Phys. Stud. 22, 69–80 (2000; Zbl 0979.53098)].

MSC:
53D55 Deformation quantization, star products
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