Is the following concept already known, perhaps with a different name?

Let $R$ be a commutative ring and $* : R \to R$ be an involutive homomorphism (the most typical case is $R=\mathbb{C}$ and $* = $ complex conjugation). A $\star$-module is an $R$-module $M$ equipped with an involutive homomorphism $* : A \to A$, where $A$ denotes the underlying abelian group of $M$, such that $(r \cdot a)^* = r^* \cdot a^*$ holds for all $r \in R$ and $a \in A$. This may also be described as a module over the twisted polynomial ring $R[T]_*$, in which $T r = r^* T$ (does this ring have a special name?). The tensor product over $R$ (!) makes the category of $*$-modules symmetric monoidal. The involution is defined by $(a \otimes b)^* = a^* \otimes b^*$.

Initially I wanted to represent the category of $*$-algebras as the category of algebras of a natural symmetric monoidal category, but the above candidate does not work: Algebras in that category satisfy $(a \cdot b)^* = a^* \cdot b^*$, but we would like to have $(a \cdot b)^* = b^* \cdot a^*$. Any ideas how to repair this?

  • $\begingroup$ I think that is just not what star-algebras are. Star-algebras are (enriched) dagger categories with one object (en.wikipedia.org/wiki/Dagger_category). $\endgroup$ – Qiaochu Yuan Sep 22 '15 at 15:56
  • $\begingroup$ @QiaochuYuan: Can you prove that star-algebras are not the algebras of a monoidal category? $\endgroup$ – Martin Brandenburg Sep 23 '15 at 10:05
  • $\begingroup$ Well, trivially, they are the algebras in the cocartesian monoidal category of $\ast$-algebras... $\endgroup$ – Zhen Lin Sep 24 '15 at 12:53
  • $\begingroup$ Yes, I meant "nontrivially". $\endgroup$ – Martin Brandenburg Sep 24 '15 at 16:49

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