Is there an explicit equivalence of categories between real and complex commutative unital C* algebras Gelfand duality shows that the category of commutative unital C* algebras is dual to the category of compact Hausdorff spaces.  This holds for the real case (see Stone spaces of Johnstone) as well as the much better known case of complex C* algebras though the proofs are not the same.  Since these two categories are both dual to the same category they should be equivalent to each other.  Is this equivalence of categories explicitly described anywhere?

I am not sure about a reference, but the equivalence is easy enough to describe:
From complex to real:
$$
  A \to  A_{sa} = \{a\in A: a^*=a\}.
  $$
From real to complex:
$$
  A \to  {\mathbb C}\otimes A.
  $$
