  The manuscript is devoted to the construction of $\Psi^\star$- algebras
containing the Bergman projection on the unit ball $B_n$ of $\mathbb C^n$. We
consider the $C^\star$-algebra $\mathcal{L}(L^2(B_n))$ of bounded operator
acting on the Hilbert space of square integrable functions on $B_n$ with
respect to the standard probability measure. We search for spectral invariant
Fr\'{e}chet subalgebras of $\mathcal{L}(L^2(B_n))$ containing the Bergman
projection $P$ which is defined on $L^2(B_n)$ with values in the space of
holomorphic functions. We use the commutator method as introduced by Gramsch.
This method generalizes the work of Beals for the spectral invariance of
pseudodifferential operators. One need not to work on the H\"{o}rmander classes
but can describe the algebra of pseudodifferential operators as continuous
commutator between scales of Sobolev spaces. To connect the analytical
properties to the geometry of the unit ball, we search for linear tangent
vector fields $X$ on the the unit sphere $\partial B_n$ so that the commutator
$[X,P]$ has a continuous extension to $L^2(B_n)$. In comparison to the work of
Bauer in for the case of the Fischer-Fock space it turns out that the results
are completely different. Our first contribution states that the set of vector
fields which provides the $\Psi^\star$-algebra for the Fock space fails even to
provide a continuity with the Bergman projection in the case of the unit ball.
Our second contribution, is the construction of vector fields being tangent to
the unit sphere which provides a new $\Psi^\star$-algebra containing the
Bergman projection of the unit ball. We prove that every linear vector field on
the unit sphere commutes with the Bergman projection on the unit ball.

Functional Analysis;Operator Algebras