Discrete quantum groups
On a discrete group \(G\) consider the algebra \(A\) of complex functions with finite supports. Multiplication in \(G\) induces a comultiplication \(\Delta\) for \(A\) by \((\Delta f)(p,q)=f(pq)\), \(f\in A\), \(p\in G\), \(q\in G\). In case \(G\) is infinite, \(\Delta(f)(g \otimes 1)\), \(\Delta(f)(1\otimes g)\in A\otimes A\) for all \(f,g\in A\); \((A,\Delta)\) is a multiplier Hopf algebra, actually a multiplier Hopf \(*\)-algebra with respect to the involution defined by \(f^*(p)=\overline{f(p)}\), \(f\in A\), \(p\in G\). The author then calls \((A,\Delta)\) a discrete quantum group if it is the direct sum of full complex matrix algebras. Let \(\iota\) denote the identity operator. Then a linear functional \(\varphi\) on \(A\) is called left invariant if \((\iota\otimes \varphi) \Delta(a)= \varphi(a)1\) for all \(a\in A\). The existence of a unique left invariant functional on \(A\) such that \(\varphi(1)=1\) is established; \(\varphi\) is positive and faithful, a left Haar measure. Analogously, a unique right Haar measure is obtained.
scientific article; zbMATH DE number 869786