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uniform continuity over locally compact quantum groupoids (Topic)

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, $\nu$, $(\mathcal G,\nu):= ([{\mathbb{G}}, G_2, \mu], \nu)$, or as derived from a (non-commutative) weak Hopf algebra (WHA), with the additional condition of uniform continuity over ${\mathbb{G}}$ defined as follows . Let us also consider a space $LUC({\mathbb{G}})$ of left uniformly continuous elements in $L^{\infty}({\mathbb{G}})$ defined over $G_2$, which is endowed with the induced product topology from the subset $G^2$ of composable pairs in the topological groupoid ${\mathbb{G}}$. 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.

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. [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.

Bibliography

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



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See Also: compact quantum groupoids, locally compact groupoid

Also defines:  uniform continuity over quantum groupoids, locally compact quantum groupoid
Keywords:  uniform continuity, locally compact quantum groupoids

Cross-references: groupoid C*-convolution algebra, C*-algebra, system, operator, group, functions, locally compact quantum group, quantum groupoids, topological groupoid, weak Hopf algebra, non-commutative, Haar system, locally compact groupoids
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This is version 5 of uniform continuity over locally compact quantum groupoids, born on 2009-04-19, modified 2009-04-19.
Object id is 677, canonical name is UcLocallyCompactQuantumGroupoids.
Accessed 783 times total.

Classification:
Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
 03. (Quantum mechanics, field theories, and special relativity )
 03.65.Fd (Algebraic methods )
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