Toeplitz operators on the Fock space with quasi-radial symbols

The Fock space $\mathcal{F}(\mathbb{C}^n)$ is the space of holomorphic functions on $\mathbb{C}^n$ that are square-integrable with respect to the Gaussian measure on $\mathbb{C}^n$. This space plays an important role in several subfields of analysis and representation theory. In particular, it has for a long time been a model to study Toeplitz operators. Esmeral and Maximenko showed in 2016 that radial Toeplitz operators on $\mathcal{F}(\mathbb{C})$ generate a commutative $C^*$-algebra which is isometrically isomorphic to the $C^*$-algebra $C_{b,u}(\mathbb{N}_0,\rho_1)$. In this article, we extend the result to $k$-quasi-radial symbols acting on the Fock space $\mathcal{F}(\mathbb{C}^n)$. We calculate the spectra of the said Toeplitz operators and show that the set of all eigenvalue functions is dense in the $C^*$-algebra $C_{b,u}(\mathbb{N}_0^k,\rho_k)$ of bounded functions on $\mathbb{N}_0^k$ which are uniformly continuous with respect to the square-root metric. In fact, the $C^*$-algebra generated by Toeplitz operators with quasi-radial symbols is $C_{b,u}(\mathbb{N}_0^k,\rho_k)$.