$R$-matrices and the Yang-Baxter equation on GNS representations of C${}^{\ast }$-bialgebras

A new construction method of $R$-matrix is given. Let $A$ be a C${}^{\ast }$-bialgebra with a comultiplication $\mathrm{\Delta }$. For two states $\omega$ and $\psi$ of $A$ which satisfy certain conditions, we construct a unitary $R$-matrix $R(\omega ,\psi )$ of the C${}^{\ast }$-bialgebra $(A,\mathrm{\Delta })$ on the tensor product of GNS representation spaces associated with $\omega$ and $\psi$. The set $\{R(\omega ,\psi ):\omega ,\psi \}$ satisfies a kind of Yang-Baxter equation. Furthermore, we show a nontrivial example of such $R$-matrices for a non-quasi-cocommutative C${}^{\ast }$-bialgebra.