We consider the following simple fact about matrices. Then we try to generalize it in the context of smooth manifolds;

Let $L$ be the collection of all $n \times n$ real matrices $A=(a_{ij})$ with the following property:

$$\sum_{i=1}^{n} a_{ij}=0$$ for every fixed $j$.

Obviousely $L$ is a Lie algebra.(As I have already learned from Qiaochu Yuan in another MO post)

Moreover the linear map $X \mapsto AX$ has non trivial kernel.

These simple facts can be modelized in an infinite dimensional manner.(note that a vector $X\in\mathbb{R}^{n}$ can be considered as a function on a finite ($n$ pointed)set $M$ equiped with discrete counting measure and a matrix is a function on $M\times M$. Now the product $AX$ has an integral representation if we replace $\sum$ by the integral sign. That is we read the expersion $\sum a_{ij}x_{j}$ in the integral form $\int_{M} a_{ij}x_{j}$ where the integration is based on the normalized counting measure.

Now we state our questions as generalization of the above simple fact about matrices.

Assume that $M$ is a compact orientable manifold or a Lie or topological group. So $M$ has a natural measure, correspond to volum form or the invariant metric or Haar measure. Assume that $g: M \times M \to \mathbb{R}$ is a smooth function which satisfies $$\int_{M} g(x,y)dx=0\;\;\;\;\;(1) $$ for all $y \in M$.

Does the linear map $A$ on $C^{\infty} (M)$ has nontrivial kernel? $$A(f)(x)=\int_{M} g(x,y)f(y)dy$$

Note that for topological groups we consider continuous functions, since smoothness is meaningles.

For our next question, we assume that $M$ is a symplectic manifold, so $M \times M$ has a natural symplectic structure. Let $L$ be the space of all smooth functions on $M\times M$ which satisfy the equation (1).

Is $L$ closed under Poisson bracket?