Finite-dimensional Hopf C*-bimodules and C*-pseudo-multiplicative unitaries

Finite quantum groupoids can be described in many equivalent ways: In terms of the weak Hopf C*-algebras of B"ohm, Nill, and Szlach'anyi or the finite-dimensional Hopf-von Neumann bimodules of Vallin, and in terms of finite-dimensional multiplicative partial isometries or the finite-dimensional pseudo-multiplicative unitaries of Vallin. In this short note, we show that in finite dimensions, the notions of a Hopf-von Neumann bimodule and of a pseudo-multiplicative unitary coincide with the notions of a concrete Hopf-C*-bimodule and of a C*-pseudo-multiplicative unitary, respectively.