Remark

The mapping $f\mapsto f(x)$ is isometric; its image is the unital star-subalgebra B generated by $x$, which is contained in the bicommutant of $x$.

If $f$ is the restriction to Sp$_A(x)$ of a function of the form $z\mapsto$ $P(z, z)$, where $P\in \mathbf{C}[X,Y]$ is a polynomial, then one has $f(x) = P(x, x^*)$ in the usual algebraic sense.