R-Matrices and the Yang-Baxter equation on GNS representations on C∗-bialgebras

A new construction method of R-matrix is given. Let A be a C∗-bialgebra with a comultiplication Δ without the assumption of the quasi-cocommutativity. For two states ω and ψ of A which satisfy certain conditions, we construct a unitary R-matrix R(ω,ψ) of the C∗-bialgebra (A,Δ) on the tensor product of GNS representation spaces associated with ω and ψ. The set {R(ω,ψ):ω,ψ} satisfies a kind of Yang-Baxter equation. Furthermore, we show a nontrivial example of such R-matrices for a non-quasi-cocommutative C∗-bialgebra.