We prove the existence of commutative C * -algebras of Toeplitz operators on every weighted Bergman space over the complex projective space P n (C). The symbols that define our algebras are those that depend only on the radial part of the homogeneous coordinates. The algebras presented have an associated pair of Lagrangian foliations with distinguished geometric properties and are closely related to the geometry of P n (C). Mathematics Subject Classification (2010). Primary 47B35;Secondary 32A36, 32M15, 53C12.

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We prove that the quasi-homogenous symbols on the projective space P n (C) yield commutative algebras of Toeplitz operators on all weighted Bergman spaces, thus extending to this compact case known results for the unit ball B n . These algebras are Banach but not C * . We prove the existence of a strong link between such symbols and algebras with the geometry of P n (C).1991 Mathematics Subject Classification. Primary 47B35; Secondary 32A36, 32M15, 53C12.

Abstract. On the weakly pseudo-convex domains Ω n p we introduce quasihomogeneous quasi-radial symbols. These are used to prove the existence of a commutative Banach algebra of Toeplitz operators on Bergman space of Ω n p . We also show that group theoretic and geometric properties for our symbols are satisfied. The results presented here contain the geometric description of the symbols introduced by N. Vasilevski in [12] for the unit ball B n .

We describe the C⁎-algebra generated by the Toeplitz operators acting on each poly-Fock space of the complex plane C with the Gaussian measure, where the symbols are bounded functions depending only on x=Re  z and have limit values at y=-∞ and y=∞. The C⁎ algebra generated with this kind of symbols is isomorphic to the C⁎-algebra functions on extended reals with values on the matrices of dimension n×n, and the limits at y=-∞ and y=∞ are scalar multiples of the identity matrix.