The exponential map is not open on the von Neumann algebra of all bounded linear operators on an infinite dimensional separable complex Hilbert space.  A [1987 article by Conway and Morrel][1] shows that the spectrum of an element of the interior of the image of the exponential map does not separate 0 from infinity.  On the other hand, a bilateral shift *U* has spectrum equal to the unit circle centered at the origin, and every Borel logarithm on the circle applied to the unitary operator $U$ yields a preimage point for *U* under the exponential map.  Hence *U* is in the image but not in the interior of the image.  

I learned about Conway and Morrel's article from [this answer][2] by David Speyer.


  [1]: http://www.ams.org/mathscinet-getitem?mr=870760
  [2]: https://mathoverflow.net/questions/154/can-you-describe-the-image-of-the-exponential-map-bh-bh/342#342