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 [[../QuantumOperatorAlgebra5/|Haar system]]}, $\nu$, Failed to parse (unknown function "\G"): {\displaystyle (\G,\nu):= ([\grp, G_2, \mu], \nu)} , or as derived from a (\htmladdnormallink{non-commutative {http://planetphysics.us/encyclopedia/AbelianCategory3.html}) [[../WeakHopfAlgebra/|weak Hopf algebra]]} (WHA), with the additional condition of uniform continuity over Failed to parse (unknown function "\grp"): {\displaystyle \grp} defined as follows . Let us also consider a space Failed to parse (unknown function "\grp"): {\displaystyle LUC(\grp)} of left uniformly continuous elements in Failed to parse (unknown function "\grp"): {\displaystyle 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 [[../EquivalenceRelation/|topological groupoid]] Failed to parse (unknown function "\grp"): {\displaystyle \grp} . This step completes the construction of uniform continuity over $LCQGn$ that can be then compared with the results obtained from `[[../WeakHopfAlgebra/|quantum groupoids]]' derived from a weak Hopf algebra.

C*-algebra Comparison and Example

Consider $LCG$ to be a [[../LocallyCompactQuantumGroup/|locally compact quantum group]]. Then consider the space $LUC(G)$ of left uniformly continuous elements in $L^{\mathrm{\infty }}(G)$ introduced in ref. . (The definition according to V. Runde (\em loc. cit.) covers both the space of left uniformly continuous [[../Bijective/|functions]] on a locally compact [[../TrivialGroupoid/|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} [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|system]] containing the [[../VonNeumannAlgebra2/|C*-algebra]] $C_o(G)$}. One may compare the [[../GroupoidCConvolutionAlgebra/|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.