Fixed point and spectral characterization of finite dimensional C-algebras

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.