Tohn F—ez`˜—ezdg—l—xyfu™rfedub Wrotex

De ne a C* algebra A over the quaternions H as follows: First, A should be an algebra in the usual sense over the reals R, and A should be a left and right vector space over H, where the di erent senses of scalar multiplication by a real agree. Then extend associativity by h(ab) = (ha)b, (ah)b = a(hb), (ab)h = a(bh), and (ha)i = h(ai), where (as always here) a, b ∈ A and h, i ∈ H. Let ∗ be an involution on A, with (a+ b) = a∗ + b∗, (ab) = b∗a∗, (ha) = a∗h, and (ah) = ha∗, where h for h ∈ H is the standard conjugation. Let ‖·‖ be a function from A to the nonnegative reals. Insist ‖a+ b‖ ≤ ‖a‖+ ‖b‖ and the same for multiplication. If ‖a‖ = 0 implies a = 0, d(a, b) := ‖a− b‖ is a metric. Insist that this is the case and that the metric is complete. Require ‖ha‖ = ‖h‖‖a‖, ‖a∗‖ = ‖a‖, and ‖a∗a‖ = ‖a‖2. I don t think this is always included in C* algebras, but let s say there is a multiplicative identity 1, with h1 = 1h and ‖1‖ = 1. I think that s all. Then H itself is a C* algebra over the quaternions.