20 Group theory and generalizations -- 20G Linear algebraic groups and related topics 20G42 Quantum groups
46 Functional analysis -- 46L Selfadjoint operator algebras 46L85 Noncommutative topology 46L89 Other 'noncommutative'' mathematics based on $C^*$-algebra theory
In this paper, the authors study the structures and some judgment criteria of a compact quantum group on type-I $C^{*}$-algebras by using results from [S.~V. Neshveyev and L. Tuset, {\it Compact quantum groups and their representation categories}, Cours Spéc., 20, Soc. Math. France, Paris, 2013; MR3204665; E. Bédos, G.~J. Murphy and L. Tuset, J. Geom. Phys. {\bf 40} (2001), no.~2, 130--153; MR1862084; Int. J. Math. Math. Sci. {\bf 31} (2002), no.~10, 577--601; MR1931751; M. Caspers and A.~G. Skalski, Bull. Lond. Math. Soc. {\bf 51} (2019), no.~4, 691--704; MR3990385]. The main conclusion of the paper under review is that if $G$ is a compact quantum group such that a unital $C^{*}$-algebra $A=C(G)$ fits into an exact sequence as stated in Equation 3.1 of this paper, then $A$ is finite-dimensional.