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

Abstract

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.