  We show that the following conditions on a C*-algebra are equivalent: (i) it
has the fixed point property for nonexpansive mappings, (ii) the spectrum of
every self adjoint element is finite, (iii) it is finite dimensional. We prove
that (i) implies (ii) using constructions given by Goebel, that (ii) implies
(iii) using projection operator properties derived from the spectral and
Gelfand-Naimark-Segal theorems, and observe that (iii) implies (i) by Brouwer's
fixed point theorem.
