Let $A$ be a $C^*$-algebra or some norm-closed algebra of operators on a Hilbert space.

In the old paper

Hille, E. On Roots and Logarithms of Elements of a Complex Banach Algebra, Math. Annalen, Bd. 136, S. 46--.57 (1958)

the question is studied, which elements $x \in A$ have the property that the exponential function is open at $x$ (i.e. every neighborhood of $x$ maps to a neighborhood of $\exp(x)$). Under some conditions on the spectrum of $x$, Hille shows that the answer is positive.

Question: Is there an example of a unital $C^*$-algebra (or operator algebra) with the property that the exponential function is not open?

More specifically: Is the exponential function open for the algebra of upper triangular operators on $\ell^2 {\mathbb N}$?

Is there an example of a Banach algebra where the exponential function is not open?