$C^{\ast }$-Algebra generated by $n$-orthogonal projections and its relationship to Toeplitz operators acting on the poly-Fock space

Starting from the system of orthogonal projections mapping the $n$-poly-Fock space $F_n^2(\mathbb{C})$ onto the true-poly-Fock components, we constructed a system of all-but-one orthogonal projections in generic position. We described the $C^{\ast }$-algebra generated by these projections via an isomorphism with a subalgebra of matrix-valued continuous functions on the two-point compactification of the real line. In addition, we studied the $C^{\ast }$-algebra generated by a Toeplitz operator with a horizontal symbol and the orthogonal projections from $F_n^2(\mathbb{C})$ onto the true-poly-Fock subspaces. An interesting fact in this work was that the $C^{\ast }$-algebras studied herein contain the $C^{\ast }$-algebra generated by all Toeplitz operators acting on $F_n^2(\mathbb{C})$ with horizontal symbols having limit values at $\pm \mathrm{\infty }$. Our approach combines unitary equivalences induced by Bargmann-type transforms with a noncommutative Stone–Weierstrass-type argument to describe the algebras.