Non-existence of universal $R$-matrix for some C$^{*}$-bialgebras

Abstract

For a C$^{*}$-bialgebra $A$ with a comultiplication $\Delta$, a universal $R$-matrix of $(A,\Delta)$ is defined as a unitary element in the multiplier algebra $M(A\otimes A)$ of $A\otimes A$ which is an intertwiner between $\Delta$ and its opposite comultiplication $\Delta^{op}$. We show that there exists no universal $R$-matrix for some C$^{*}$-bialgebras.