The Early History of the Ham Sandwich Theorem

Proof Let Sym(A) := {a A : a* = a), and let a belong to Sym(A). Consider the subalgebra B of A that comprises all polynomials in a with real coefficients. Then B is contained in Sym(A). Hence B satisfies hypotheses of Theorem 3.1. Since B is also commutative, B is isomorphic to R or C. Hence a = X1l for some real or complex number ). This shows that Sym(A) itself is isomorphic to R or C. Now the conclusion follows from Lemma 2.1 of [1]. 0