###Question

The question asked is:

> On a manifold $M$ equipped with a Riemann metric $g$ and a symplectic structure $\omega$, with $\ast$ the Hodge star and $\ast_s$ the symplectic star, does $\ast=\ast_s$ iff $(M,g,\omega)$ is Kähler?

### Definitions 

Here the star operators $\ast$ and $\ast_s$ are defined in the usual way (see for example, equation 3.9 of Tseng and Yau *[Cohomology and Hodge Theory on Symplectic Manifolds: II][Tseng/Yau]*):

$$ A\wedge \ast B := \langle A,B\rangle_{g^{-1}} dV_g$$ 
$$A\wedge \ast_{s} B := \langle A,B\rangle_{\omega^{-1}} dV_\omega $$

where $A$ and $B$ are $k$-forms, $\langle\cdot,\cdot\rangle_g$ is the inner product associated to $g$, $dV_g$ is the volume form associated to $g$, and $\langle\cdot,\cdot\rangle_\omega$ and $dV_\omega$ are defined analogously with respect to $\omega$.

### Motivation

In seeking a mathematically natural link between Hilbert-space expositions of quantum mechanics and [product space expositions][Verstraete], both the Hodge star operator and the symplectic star operator enter naturally (for example, in the context of [Onsager theory][Onsager]). Moreover, in cases of practical quantum systems engineering interest it is commonly observed that: 

* the two star operations are identical, and 
* the dynamical state-manifold is Kählerian.

Does each of these observations mathematically imply the other?  An engineer-friendly reference for this fact (if it is a fact) would be very welcome---it's not easy to find discussion of the practical relevance of the symplectic star operation.


[Tseng/Yau]: http://arxiv.org/abs/1011.1250 (review by Li-Sheng Tseng and Shing-Tung Yau)

[Verstraete]: http://arxiv.org/abs/0907.2796 (review of matrix product states)

[Onsager]: http://faculty.washington.edu/sidles/ENC_2011/Onsager_transport.pdf (summary of pullback dynamics)