Convolution of Trace Class Operators over Locally Compact Quantum Groups
| Autor: | Matthias Neufang, Zhiguo Hu, Zhong Jin Ruan |
|---|---|
| Rok vydání: | 2013 |
| Předmět: |
Discrete mathematics
Quantum group General Mathematics 010102 general mathematics Locally compact group 01 natural sciences C*-algebra Multiplier (Fourier analysis) Compact space 0103 physical sciences Noncommutative harmonic analysis Bimodule 010307 mathematical physics Locally compact space 0101 mathematics Mathematics |
| Zdroj: | Canadian Journal of Mathematics. 65:1043-1072 |
| ISSN: | 1496-4279 0008-414X |
| DOI: | 10.4153/cjm-2012-030-5 |
| Popis: | We study locally compact quantum groups 𝔾 through the convolution algebras L1(𝔾) and (T(L2(𝔾)); ⊳). We prove that the reduced quantum groupC*-algebraC0(𝔾) can be recovered fromthe convolution ⊳ by showing that the right T(L2(𝔾))-module 〈K(L2(𝔾)) ⊳ T(L2(𝔾))〉 is equal to C0(𝔾). On the other hand, we show that the left T(L2(𝔾)(-module 〈T(L2(𝔾)) ⊳ K(L2(𝔾))〉 is isomorphic to the reduced crossed product C0(Ĝ ) r⋉C0(𝔾), and hence is a much larger C*-subalgebra of B(L2(𝔾)).We establish a natural isomorphism between the completely bounded right multiplier algebras of L1(𝔾) and (T(L2(𝔾)); ⊳), and settle two invariance problems associated with the representation theorem of Junge–Neufang–Ruan (2009). We characterize regularity and discreteness of the quantum group 𝔾 in terms of continuity properties of the convolution . on T(L2(𝔾))⊳and settle two invariance problems associated with the representation theorem of Junge–Neufang–Ruan (2009). We characterize regularity and discreteness of the quantum group 𝔾 in terms of continuity properties of the convolution ⊳ on T(L2(𝔾)). We prove that if 𝔾 is semiregular, then the space 〈T(L2(𝔾)) ⊳ B(L2(𝔾))〉 of right 𝔾-continuous operators on L2(𝔾), which was introduced by Bekka (1990) for L1(𝔾), is a unital C*-subalgebra of B(L2(𝔾)). In the representation framework formulated by Neufang–Ruan–Spronk (2008) and Junge–Neufang–Ruan, we show that the dual properties of compactness and discreteness can be characterized simultaneously via automatic normality of quantum group bimodule maps on B(L2(𝔾)). We also characterize some commutation relations of completely bounded multipliers of (T(L2(𝔾)); ⊳) over B(L2(𝔾)). |
| Databáze: | OpenAIRE |
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