This is an announcement for the paper "The fixed point property for a class of nonexpansive maps in L\sp\infty(\Omega,\Sigma,\mu)" by Cleon S. Barroso. Abstract: For a finite and positive measure space $(\Omega,\Sigma,\mu)$ and any weakly compact convex subset of $L\sp\infty(\Omega,\Sigma,mu)$, a fixed point theorem for a class of nonexpansive self-mappings is proved. An analogous result is obtained for the space $C(\Omega)$. An illustrative example is given. Archive classification: Functional Analysis Mathematics Subject Classification: 47H10 Remarks: 4 pages The source file(s), Cleonfp.tex: 11461 bytes, is(are) stored in gzipped form as 0404235.gz with size 4kb. The corresponding postcript file has gzipped size 32kb. Submitted from: cleonbar@mat.ufc.br The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0404235 or http://arXiv.org/abs/math.FA/0404235 or by email in unzipped form by transmitting an empty message with subject line uget 0404235 or in gzipped form by using subject line get 0404235 to: math@arXiv.org.