Normal essential eigenvalues in the boundary of the numerical range

A purely geometric property of a point in the boundary of the numerical range of an operator $T$ on Hilbert space is examined which implies that such a point is the value at $T$ of a multiplicative linear functional of the $C^*$-algebra, $C^*(T)$, generated by $T$ and the identity operator. Roughly speaking, such a property means that the boundary of the numerical range (of $T$) has infinite curvature at that point. Furthermore, it is shown that if such a point is not a sharp linear corner of the numerical range of $T$, then the multiplicative linear functional vanishes on the compact operators in $C^*(T)$.