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# uniform continuity over locally compact quantum groupoids

# 0.1 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, $\nu$, $(\mathcal{G},\nu):=([{\mathbb{G}},G_{2},\mu],\nu)$, or as derived from a (non-commutative) weak Hopf algebra (WHA), with the additional condition of *uniform continuity* over ${\mathbb{G}}$ defined as follows . Let us also consider a space $LUC({\mathbb{G}})$ of left uniformly continuous elements in $L^{{\infty}}({\mathbb{G}})$ defined over $G_{2}$, which is endowed with the induced product topology from the subset $G^{2}$ of composable pairs in the *topological groupoid* ${\mathbb{G}}$. 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.

# 0.1.1 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.

# References

- 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).

## Mathematics Subject Classification

46L05*no label found*81R50

*no label found*46M20

*no label found*18B40

*no label found*22A22

*no label found*

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