PlanetPhysics/Uc Locally Compact Quantum Groupoids

Uniform Continuity over Locally Compact Quantum Groupoids
Let us consider locally compact quantum groupoids ($$LCQGn$$) defined as \htmladdnormallink{locally compact groupoids {http://planetphysics.us/encyclopedia/LocallyCompactGroupoid.html} endowed with a Haar system}, $$\nu$$, $$(\G,\nu):= ([\grp, G_2, \mu], \nu)$$, or as derived from a (\htmladdnormallink{non-commutative {http://planetphysics.us/encyclopedia/AbelianCategory3.html}) 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. . (The definition according to V. Runde (\em 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 \htmladdnormallink{operator {http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} 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.