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Boundary representations on co-invariant subspaces of Bergman space

Let M be an invariant subspace of the Bergman space L 2 a (D) and S M be the compression of the coordinate multiplication operator M z to the co-invariant subspace L 2 a (D) ⊖ M. The present paper determines when the identity representation of C* (S M ) is a boundary representation for the Banach subalgebra B(S M ). The paper also considers boundary representations on the co-invariant subspaces of L 2 a (B n ).