$ C^* $-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^* $-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^* $-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^* $-algebras studied herein contain the $ C^* $-algebra generated by all Toeplitz operators acting on $ F_n^2(\mathbb{C}) $ with horizontal symbols having limit values at $ \pm \infty $. Our approach combines unitary equivalences induced by Bargmann-type transforms with a noncommutative Stone–Weierstrass-type argument to describe the algebras.