A real analogue of the Gel\cprime fand-Neumark theorem

Abstract

Let A be a real Banach *-algebra enjoying the following three conditions: ||x*x||=||x*| ||x||, Spx*x>O, and ||x*|| =||x| (xEA). It is shown, after Ingelstam, Palmer, and Behncke, as a real analogue of the Gelfand-Neumark theorem, that A is isometrically *-isomorphic onto a real C*-algebra acting on a suitable real (or complex) Hilbert space. The converse is obvious. The aim of this note is, as a real analogue of the Gelfand-Neumark theorem, to prove the following THEOREM. A real Banach *-algebra A is isometrically *-isomorphic onto a real C*-algebra acting on a real (or complex) Hilbert space if and only if it satisfies the following three conditions: (1) H1x*xH1 = !1x*H1 lHxHi, (2) Spx*x_O,and (3) ||x*|| = l|x|| (xEA). The above theorem was conjectured explicitly by Rickart [5, p. 181] and proved by Ingelstam [2] (cf. also Palmer [3], [4] and Behncke [1 ]). Their proofs were based on complexification of a real Banach *-algebra. An alternative proof which we shall give in this note will be done by real *-representation on real Hilbert space and by complexification of a real Hilbert space. Let A be a real Banach *-algebra satisfying the conditions stated in the theorem, and H the set of hermitian elements in A. Let R be the field of real numbers. In view of (2), the involution is hermitian. Put y(h) = sup(X; X a spectrum of h) for all h in H. In view of (2), A is symmetric. In view of (3), the involution is continuous. So, we can make use of Rickart [5, Lemma 4.7.10] to get the sublinearity of jA on H, that is, (i) y(ah)=apt(h) and Received by the editors July 11, 1969. A MS Subject Classifications. Primary 4660, 4665.