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What references are there for studying finite-dimensional $*$-algebras over the field $\mathbb R$ in their full generality? We assume these are associative and unital.

Note that:

  • Not every algebra over $\mathbb R$ can be made into a $*$-algebra. The path algebra of the quiver $\bullet\rightarrow\bullet\leftarrow\bullet$ is a counterexample. In fact, the path algebras of most quivers provide counterexamples.

  • Not every $*$-algebra is a $C^*$-algebra.

  • Many finite-dimensional $\mathbb R$-algebras of practical interest can be made into $*$-algebras. For instance: commutative algebras, group algebras, Clifford algebras, and matrix algebras over any of the preceeding.

The study of these things seems rather non-trivial, so a reference would be appreciated.

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    $\begingroup$ The path algebras of quivers ARE very much of practical interest, though :-) $\endgroup$
    – M.G.
    Commented Jul 1, 2023 at 18:39
  • $\begingroup$ @M.G. Sure. I've edited the question. $\endgroup$
    – wlad
    Commented Jul 2, 2023 at 10:31
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    $\begingroup$ Chapter 2 of "Schmüdgen, Konrad. An invitation to unbounded representations of*-algebras on Hilbert space. Cham: Springer, 2020." covers a lot of material on *-algebras over R, properties that do/do-not transfer through their complexification, etc. $\endgroup$
    – Greg Zitelli
    Commented Jul 3, 2023 at 16:35

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