Fixed point properties of -algebras

This paper derives relations between the following properties of a C ∗ -algebra: (i) it has the fpp, (ii) the spectrum of every self-adjoint element is finite, (iii) it is finite dimensional, (iv) it is generated by two projections p and q and the spectrum of p + q is homeomorphic to a compact ordinal α < ω ω , (v) it is generated by two projections and the real Banach algebra generated by every self-adjoint element has the w-fpp, (vi) it has the w-fpp. We prove that (i) implies (ii) using standard fixed point theory, give two proofs that (ii) implies (iii), one based on a result of Ogasawara and another based on geometric properties of projections, and observe that (iii) implies (i) by Brouwerʹs fixed point theorem. We prove that (iv) implies (v) using the structure of the universal C ∗ -algebra generated by two projections, and discuss a conjecture that ensures (iv) implies (vi).