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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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##### References:
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