Uniform Continuity over Locally Compact Quantum Groupoids

Let us consider Locally Compact Quantum Groupoids ($LCQGn$ ) defined as locally compact groupoids endowed with a Haar system, $\nu$ , $(\G,\nu):= ([\grp, G_2, \mu], \nu)$ , or as derived from a (non-commutative) weak Hopf algebra (WHA), with the additional condition of uniform continuity over $\grp$ defined as follows . Let us also consider a space $LUC(\grp)$ of left uniformly continuous elements in $L^{\infty}(\grp)$ defined over $G_2$ , which is endowed with the induced product topology from the subset $G^2$ of composable pairs in the topological groupoid $\grp$ . This step completes the construction of uniform continuity over $LCQGn$ that can be then compared with the results obtained from `quantum groupoids' derived from a weak Hopf algebra.

C*-algebra Comparison and Example

Consider $LCG$ to be a locally compact quantum group. Then consider the space $LUC(G)$ of left uniformly continuous elements in $L^{\infty}(G)$ introduced in ref. [2]. (The definition according to V. Runde (loc. cit.) covers both the space of left uniformly continuous functions on a locally compact group and (Granirer's) uniformly continuous functionals on the Fourier algebra.) Also consider $LUC(G)$ which is then an operator system containing the C*-algebra $C_o(G)$ . One may compare the groupoid C*-convolution algebra, $G_{CA}$ - obtained in the general case- with the C*-algebra $C_o(G)$ obtained from $LUC(G)$ in the particular case of uniform continuity over a locally compact group.

Bibliography

1

M. Buneci. 2003. Groupoid Representations, Publs: Ed. Mirton, Timishoara.

2

V. Runde. 2008. Uniform continuity over locally compact quantum groups. (math.OA -arxiv/0802.2053v4).