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

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

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