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Journal of the London Mathematical Society 2001 64(1):144-160; doi:10.1017/S0024610701002174
© 2001 by London Mathematical Society
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© The London Mathematical Society

Direct Sums of Operator Spaces

Timur Oikhberg

Department of Mathematics, University of Texas Austin, TX 78712, USA, timur{at}math.utexas.edu

Received 10 July 2000.

It is proved that if X and Y are operator spaces such that every completely bounded operator from X into Y is completely compact and Z is a completely complemented subspace of X {oplus} Y, then there exists a completely bounded automorphism {tau}: X {oplus} Y -> X {oplus} Y with completely bounded inverse such that {tau}Z = X0 {oplus} Y0, where X0 and Y0 are completely complemented subspaces of X and Y, respectively. If X and Y are homogeneous, the existence is proved of such a {tau} under a weaker assumption that any operator from X to Y is strictly singular. An upper estimate is obtained for ||{tau}||cb||{tau}–1||cb if X and Y are separable homogeneous Hilbertian operator spaces. Also proved is the uniqueness of a ‘completely unconditional’ basis in X {oplus} Y if X and Y satisfy certain conditions.


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