Banach Journal of Mathematical Analysis

A Hilbert space approach to approximate diagonals for locally compact quantum groups

Benjamin Willson

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For a locally compact quantum group $\mathbb{G}$, the quantum group algebra $L^1(\mathbb{G})$ is operator amenable if and only if it has an operator bounded approximate diagonal. It is known that if $L^1(\mathbb{G})$ is operator biflat and has a bounded approximate identity then it is operator amenable. In this paper, we consider nets in $L^2(\mathbb{G})$ which suffice to show these two conditions and combine them to make an approximate diagonal of the form $\omega_{{W'}^*\xi\otimes\eta}$ where $W$ is the multiplicative unitary and $\xi\otimes\eta$ are simple tensors in $L^2(\mathbb{G})\otimes L^2(\mathbb{G})$. Indeed, if $L^1(\mathbb{G})$ and $L^1(\hat{\mathbb{G}})$ both have a bounded approximate identity and either of the corresponding nets in $L^2(\mathbb{G})$ satisfies a condition generalizing quasicentrality then this construction generates an operator bounded approximate diagonal. In the classical group case, this provides a new method for constructing approximate diagonals emphasizing the relation between the operator amenability of the group algebra $L^1(G)$ and the Fourier algebra $A(G)$.

Article information

Banach J. Math. Anal., Volume 9, Number 3 (2015), 248-260.

First available in Project Euclid: 19 December 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 43A07: Means on groups, semigroups, etc.; amenable groups
Secondary: 20G42: Quantum groups (quantized function algebras) and their representations [See also 16T20, 17B37, 81R50] 81R50: Quantum groups and related algebraic methods [See also 16T20, 17B37] 22D35: Duality theorems

Locally compact quantum group amenability operator amenability approximate diagonal quasicentral approximate identity


Willson, Benjamin. A Hilbert space approach to approximate diagonals for locally compact quantum groups. Banach J. Math. Anal. 9 (2015), no. 3, 248--260. doi:10.15352/bjma/09-3-18.

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