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International Journal of Mathematics and Mathematical Sciences
Volume 1, Issue 2, Pages 209-215

On the range of completely bounded maps

Department of Mathematics, Wayne State University, Detroit 48202, Michigan, USA

Received 28 March 1977

Copyright © 1978 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


It is shown that if every bounded linear map from a C*-algebra α to a von Neumann algebra β is completely bounded, then either α is finite-dimensional or β𝒞Mn, where 𝒞 is a commutative von Neumann algebra and Mn is the algebra of n×n complex matrices.