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Bulletin of the London Mathematical Society Advance Access originally published online on January 6, 2009
Bulletin of the London Mathematical Society 2009 41(1):57-62; doi:10.1112/blms/bdn104
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© 2009 London Mathematical Society

Commuting holomorphic maps on the spectral unit ball

C. Costara

Département de Mathématiques et de Statistique
Université Laval
Québec
G1K 7P4
Canada

Received 19 September 2007. Revision received 21 August 2008.

We prove that if F is a holomorphic map from the open spectral unit ball of a primitive Banach algebra into itself satisfying F(0) = 0, F' (0) = I and F(x) x = xF(x) for every x, then F is the identity map. Using this, we prove that if A is a semisimple Banach algebra and B is a primitive Banach algebra, then any unital spectral isometry from A onto B which locally preserves commutativity is a Jordan morphism. The same is true when A and B are both assumed to be von Neumann algebras.

2000 Mathematics Subject Classification 46G20 (primary), 47B48, 16R50 (secondary).


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