A note on polynomial operator approximation

An example is given of an operator $T$ contained in a block-diagonal algebra of operators $\mathcal {A}$, an ideal $J \subset \mathcal {A}$ and an infinite set of polynomials $\mathcal {P}$ for which there is a $K \in J$ satisfying ${\left \| {p(T + K)} \right \|_{\mathcal {A}}} = {\left \| {p(T + K)} \right \|_{\mathcal {A}/J}}$ for any finite subset of $\mathcal {P}$ but for which there is no $K \in J$ satisfying ${\left \| {p(T + K)} \right \|_{\mathcal {A}}} = {\left \| {p(T + K)} \right \|_{\mathcal {A}/J}}$ for all $p \in \mathcal {P}$. This sheds some light on a well-known question of C. Olsen.