Boundary representations and rigidity of submodules

This paper considers two problems on the Fock type spaces F s ( 0 < s ⩽ 1 ). Firstly, it is shown that on the space F s ( 0 < s < 1 ), the identity representation of C ∗ ( I , T 1 , … , T n ) is a boundary representation for the Banach subalgebra B ( I , T 1 , … , T n ) , while on the space F 1 , it is not. Secondly, it is shown that all the submodules of F 1 are rigid.