A remark on compact matrix quantum groups (Q804682)

The author finds a simpler set of axioms, which define the matrix quantum pseudogroup. Let A be a unital \(C^*\)-algebra and u an \(N\times N\) matrix with entries belonging to A. We recall that \(G=(A,u)\) is a compact matrix pseudogroup if the following three axioms are satisfied: 1) A is the smallest \(C^*\)-algebra containing all matrix elements of u. 2) There exists a *-algebra homomorphism F: \(A\mapsto A\otimes A\) such that \(F(u_{kl})=\sum^{N}_{r=1}u_{kr}\otimes u_{rl}\) 3) There exists a linear antimultiplicative mapping k: \(B\to B\), where B is the *-subalgebra of A, generated by \(\{u_{kl}\}\), such that \(k(k(a^*)^*)=a\) for all \(a\in B\) and \(\sum k(u_{kr})u_{rl}=\delta_{kl}I\), \(\sum_{r}u_{kr}k(u_{rl})=\delta_{kl}I\), \(k,l=1,2,...,N\), where I is a unit of A. The author proves that the last axiom may be replaced by the following: 3.1) u and \(u^ T\) are invertible, where \(u^ T\) is the transpose matrix.