## 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

Zbl 0979.53098
Full Text:

### References:

 [2] Bordemann M., Ginot G., Halbout G., Herbig H.-C., Waldmann S. Star-représentations sur des sous-variétés coïsotropes. Preprint math.QA/0309321 (September 2003), 35 pp [3] Bordemann M., Neumaier N., Nowak C., Waldmann S. Deformation of Poisson brackets. Unpublished discussions on the quantization problem of general Poisson brackets, June 1997 [9] Bursztyn H., Waldmann S. (2003). Completely positive inner products and strong Morita equivalence. Preprint math.QA/0309402 (2000), To appear in Pacific J. Math · Zbl 1111.53071 [13] Nowak C.J. (1997). Über Sternprodukte auf nichtregulären Poissonmannigfaltigkeiten. PhD thesis, Fakultät fÜr Physik, Albert-Ludwigs-Universität, Freiburg, 1997
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.