The relations of the non-commutative coefficient algebra of the unitary group

Abstract

Let be na infinite-dimensional Hilbert space. Any unitary operator U on can be written as a matrix {U_pq}_o≦p,q≦d whose entries are bounded operators on . The algebra generated by the operator-valued functions is isomorphic to the complex algebra generated by the unit and the noncommutative indeterminates x_pq, x*_pq with the relations In order to prove this, the corresponding result for the commutative coefficient algebra of the unitary group u(ℂ^d) is needed, i.e. for the algebra generated by the complex-valued functions Moreover, the following result is obtained: Let F(y₁, ..., y_n) be a polynomial in the independent non-commutative indeterminates y₁, ..., y_n and assume that F(A₁ ..., A_n)=0 for all bounded operators A₁, ..., A_n on . Then F ≡ O.