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The manuscript is devoted to the construction of $\Psi^\star$- algebrascontaining the Bergman projection on the unit ball $B_n$ of $\mathbb C^n$. Weconsider the $C^\star$-algebra $\mathcal{L}(L^2(B_n))$ of bounded operatoracting on the Hilbert space of square integrable functions on $B_n$ withrespect to the standard probability measure. We search for spectral invariantFr\'{e}chet subalgebras of $\mathcal{L}(L^2(B_n))$ containing the Bergmanprojection $P$ which is defined on $L^2(B_n)$ with values in the space ofholomorphic functions. We use the commutator method as introduced by Gramsch.This method generalizes the work of Beals for the spectral invariance ofpseudodifferential operators. One need not to work on the H\"{o}rmander classesbut can describe the algebra of pseudodifferential operators as continuouscommutator between scales of Sobolev spaces. To connect the analyticalproperties to the geometry of the unit ball, we search for linear tangentvector 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 ofBauer in for the case of the Fischer-Fock space it turns out that the resultsare completely different. Our first contribution states that the set of vectorfields which provides the $\Psi^\star$-algebra for the Fock space fails even toprovide a continuity with the Bergman projection in the case of the unit ball.Our second contribution, is the construction of vector fields being tangent tothe unit sphere which provides a new $\Psi^\star$-algebra containing theBergman projection of the unit ball. We prove that every linear vector field onthe unit sphere commutes with the Bergman projection on the unit ball.

The construction of $\Psi^\star$-algebra by commutator method containing the Bergman projection on the unit ball