PlanetPhysics/Uc Locally Compact Quantum Groupoids

Uniform Continuity over Locally Compact Quantum Groupoids edit

Let us consider locally compact quantum groupoids () defined as \htmladdnormallink{locally compact groupoids {http://planetphysics.us/encyclopedia/LocallyCompactGroupoid.html} endowed with a Haar system}, , 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}) 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 , which is endowed with the induced product topology from the subset of composable pairs in the topological groupoid Failed to parse (unknown function "\grp"): {\displaystyle \grp} . This step completes the construction of uniform continuity over that can be then compared with the results obtained from `quantum groupoids' derived from a weak Hopf algebra.

C*-algebra Comparison and Example edit

Consider to be a locally compact quantum group. Then consider the space of left uniformly continuous elements in introduced in ref. [1]. (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 which is then an \htmladdnormallink{operator {http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} system containing the C*-algebra }. One may compare the groupoid C*-convolution algebra, -- obtained in the general case-- with the C*-algebra obtained from in the particular case of uniform continuity over a locally compact group.

All Sources edit

[2][1]

References edit

  1. 1.0 1.1 V. Runde. 2008. Uniform continuity over locally compact quantum groups. (math.OA -arxiv/0802.2053v4).
  2. M. Buneci. 2003. Groupoid Representations , Publs: Ed. Mirton, Timishoara.