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?

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?

EDIT: Jonas Meyer has clarified that $B(H)$ is a counterexample. It remains unclear what happens in general, or whether there is any description of the class of Banach algebras, where the exponential map is open. In particular, it remains unclear for the algebra of upper triangular operators on $\ell^2 {\mathbb N}$.