Abstract
We are given a semiprime unital ring A with * such that x*x = xx* for all elements x of A. We will show that both elements x + x* and xx* are central elements. In the case in which A is a quaternion algebra over a field F in the sense given by Albert, we show that * is unique and coincides with the canonical involution. We also provide specific constructions of quaternion division algebras A with canonical involution over a field F of one of the following types: (i) F is a function field in two variables over a ground field of unspecified characteristic; (ii) F is a function field over the Galois field GF(2n); and (iii) F is a function field over the Galois field GF(pn) where p is an odd prime number and n is a natural number.
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