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, $\nu$, $(𝒢,\nu ):=([𝔾,G_2,\mu ],\nu )$, 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 $L^{\mathrm{\infty }}(𝔾)$ defined over $G_2$, which is endowed with the induced product topology from the subset $G^2$ of composable pairs in the topological groupoid $𝔾$. 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 $L^{\mathrm{\infty }}(G)$ 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 $L\mathit{}U\mathit{}C\mathit{}\mathrm{(}G\mathrm{)}$ which is then an operator system containing the C*-algebra $C_o\mathrm{}\mathrm{(}G\mathrm{)}$. One may compare the groupoid C*-convolution algebra, $G_{C\mathit{}A}$ – obtained in the general case– with the C*-algebra (http://planetmath.org/CAlgebra3) $C_o\mathit{}\mathrm{(}G\mathrm{)}$ obtained from $L\mathit{}U\mathit{}C\mathit{}\mathrm{(}G\mathrm{)}$ in the particular case of uniform continuity over a locally compact (http://planetmath.org/LocallyCompact) group.