C*-ideals generated by polynomials

The *-algebra A(1) is defined to be the free unital *-algebra with one gene rator x. A *-ideal I of A(1) is defined to be a C*-ideal if A(1)/I may be e mbedded into a C*-algebra. It is proved that if I is a *-ideal of A(1) gene rated by polynomials in A(1), then I is a C*-ideal. This is not true for ge neral *-ideals of A(1). 1991 Mathematics subject classification: 46L.