Compact quantum group

In mathematics, a compact quantum group is an abstract structure on a unital separable C*-algebra axiomatized from those that exist on the commutative C*-algebra of 'continuous complex-valued functions' on a compact quantum group. In mathematics, a compact quantum group is an abstract structure on a unital separable C*-algebra axiomatized from those that exist on the commutative C*-algebra of 'continuous complex-valued functions' on a compact quantum group. The basic motivation for this theory comes from the following analogy. The space of complex-valued functions on a compact Hausdorff topological space forms a commutative C*-algebra. On the other hand, by the Gelfand Theorem, a commutative C*-algebra is isomorphic to the C*-algebra of continuous complex-valued functions on a compact Hausdorff topological space, and the topological space is uniquely determined by the C*-algebra up to homeomorphism. S. L. Woronowicz introduced the important concept of compact matrix quantum groups, which he initially called compact pseudogroups. Compact matrix quantum groups are abstract structures on which the 'continuous functions' on the structure are given by elements of a C*-algebra. The geometry of a compact matrix quantum group is a special case of a noncommutative geometry. For a compact topological group, G, there exists a C*-algebra homomorphism where C(G) ⊗ C(G) is the minimal C*-algebra tensor product — the completion of the algebraic tensor product of C(G) and C(G)) — such that for all f ∈ C ( G ) {displaystyle fin C(G)} , and for all x , y ∈ G {displaystyle x,yin G} , where for all f , g ∈ C ( G ) {displaystyle f,gin C(G)} and all x , y ∈ G {displaystyle x,yin G} . There also exists a linear multiplicative mapping