Similarity-invariant continuous functions on $\mathcal {L}(\mathcal {H})$

Abstract:

Let $f:\mathcal {L}(\mathcal {H}) \to X$ be a continuous function from the algebra of all bounded linear operators acting on a complex infinite dimensional Hilbert space $\mathcal {H}$ into a ${T_1}$-topological space $X$. If $f(WA{W^{ - 1}}) = f(A)$ for all $A$ in $\mathcal {L}(\mathcal {H})$ and all invertible $W$, then $f$ is a constant function. The same result is true for a function $f$ satisfying the above conditions defined on a connected open subset of $\mathcal {L}{(\mathcal {H})_0} = \{ T \in \mathcal {L}(\mathcal {H}):T \text {has no normal eigenvalues}$.