Fixed point property of direct sums

For a uniformly convex space Z, we show that Z-direct sums (X-1 circle plus ... circle plus X-N)(Z) of Banach spaces X-1,..., X-N with R(a, X-i)< 1 + a for some a is an element of (0, 1] have the fixed point property for nonexpansive mappings. As a direct consequence, the result holds for all psi-direct sums with psi being strictly convex. The same result is extended to all psi-direct sums X circle plus(psi) Y of spaces X and Y with property (M), whenever psi not equal psi(1). The permanence of properties that are sufficient for the fixed point property are obtained for Z-direct sums (and then for psi-direct sums). Such properties include the properties R(X) < 2, WNUS, C-NJ (a, X) < 2, UKK, and NUC. (C) 2005 Elsevier Ltd. All rights reserved.