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, ν, (𝒢,ν) := ([𝔾,G₂,μ],ν), or as derived from a (non-commutative) weak Hopf algebra (WHA), with the additional condition of uniform continuity over 𝔾 defined as follows . Let us also consider a space LUC(𝔾) of left uniformly continuous elements in L^∞(𝔾) defined over G ₂, which is endowed with the induced product topology from the subset G² of composable pairs in the topological groupoid 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^∞(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.

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