Real Hilbert algebras with identity

Lars Ingelstam recently proved [2] that a real Hubert algebra with identity is a division algebra and must, therefore, be isomorphic to the reals, complexes or real quaternions. This interesting result had been conjectured by I. Kaplansky. The purpose of this note is to offer a greatly simplified version of Ingelstam's proof. Our proof uses only very elementary facts about Banach algebras and is entirely selfcontained. Let A be a real Hubert algebra with identity. That is, we assume that A is a real Hubert space with inner product (x, y) which is also a Banach algebra with identity e under the norm ||x||=(x, x)in and, in addition, we have ||e|| =1. In any Banach algebra B, an element x in B has an inverse in B provided that F has an identity e and ||e — x|| <1. This remark leads to