A new duality via the Haagerup tensor product. (English) Zbl 1429.46037
The following review is a minimally augmented version of the abstract.
The authors initiate the study of a new notion of duality defined with respect to the module Haagerup tensor product. This notion not only recovers the standard operator space dual for Hilbert \(C^*\)-modules, it also captures quantum group duality in a fundamental way. The techniques adopted by the authors include a mixture of quantum group theory and the geometry of operator space tensor products.
If \(A\) is a completely contractive Banach algebra, the authors investigate a new form of duality by considering \(A^h:=(A\otimes_A^hA)^*\), where \(\otimes_A^h\) is the module Haagerup tensor product. Using this notion, the authors compute the Haagerup dual for various operator algebras arising from \(l^p\)-spaces. In particular, the authors show that the dual of \(l^1\) under any operator structure is \(\min l^\infty\).
In the setting of abstract harmonic analysis, the authors generalize a result of N. T. Varopoulos [Math. Scand. 37, 173–182 (1975; Zbl 0337.46041)] by showing that \(C(G)\) is an operator algebra under convolution for any compact Kac algebra \(G\). The authors then prove that, for a compact Kac algebra \(G\) whose dual \(\hat{G}\) is weakly amenable, the corresponding Haagerup dual \(C(G)^h\cong l^\infty (\hat{G})\) and, in particular, for any weakly amenable discrete group \(G\), \(C_\lambda^*(G)^h\cong l^\infty(G)\).
The authors initiate the study of a new notion of duality defined with respect to the module Haagerup tensor product. This notion not only recovers the standard operator space dual for Hilbert \(C^*\)-modules, it also captures quantum group duality in a fundamental way. The techniques adopted by the authors include a mixture of quantum group theory and the geometry of operator space tensor products.
If \(A\) is a completely contractive Banach algebra, the authors investigate a new form of duality by considering \(A^h:=(A\otimes_A^hA)^*\), where \(\otimes_A^h\) is the module Haagerup tensor product. Using this notion, the authors compute the Haagerup dual for various operator algebras arising from \(l^p\)-spaces. In particular, the authors show that the dual of \(l^1\) under any operator structure is \(\min l^\infty\).
In the setting of abstract harmonic analysis, the authors generalize a result of N. T. Varopoulos [Math. Scand. 37, 173–182 (1975; Zbl 0337.46041)] by showing that \(C(G)\) is an operator algebra under convolution for any compact Kac algebra \(G\). The authors then prove that, for a compact Kac algebra \(G\) whose dual \(\hat{G}\) is weakly amenable, the corresponding Haagerup dual \(C(G)^h\cong l^\infty (\hat{G})\) and, in particular, for any weakly amenable discrete group \(G\), \(C_\lambda^*(G)^h\cong l^\infty(G)\).
Reviewer: Madathum K. Viswanath (Chennai)
MSC:
| 46L07 | Operator spaces and completely bounded maps |
| 46L65 | Quantizations, deformations for selfadjoint operator algebras |
| 46L89 | Other “noncommutative” mathematics based on \(C^*\)-algebra theory |
| 43A20 | \(L^1\)-algebras on groups, semigroups, etc. |
Citations:
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