## Banach Journal of Mathematical Analysis

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

Benjamin Willson

#### Abstract

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

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

Dates
First available in Project Euclid: 19 December 2014

https://projecteuclid.org/euclid.bjma/1419001716

Digital Object Identifier
doi:10.15352/bjma/09-3-18

Mathematical Reviews number (MathSciNet)
MR3296138

Zentralblatt MATH identifier
1311.43004

#### Citation

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. https://projecteuclid.org/euclid.bjma/1419001716

#### References

• O.Y. Aristov, V. Runde and N. Spronk, Operator biflatness of the Fourier algebra and approximate indicators for subgroups, J. Funct. Anal. 209 (2004), no. 2, 367–387.
• E. Bédos and L. Tuset, Amenability and co-amenability for locally compact quantum groups, Internat. J. Math. 14 (2003), no. 8, 865–884.
• M. Caspers, H.H. Lee and É. Ricard, Operator biflatness of the $L^1$-algebras of compact quantum groups, J. Reine Angew. Math. (2013).
• M. Daws and V. Runde, Reiter's properties ($P_1$) and ($P_2$) for locally compact quantum groups, J. Math. Anal. Appl. 364 (2010), no. 2, 352 – 365.
• P. Desmedt, J. Quaegebeur and S. Vaes, Amenability and the bicrossed product construction, Illinois J. Math. 46 (2002), no. 4, 1259–1277.
• E.G. Effros and Z.-J. Ruan, Operator spaces, London Math. Soc. Monogr. New Series, vol. 23, The Clarendon Press, Oxford University Press, New York, 2000.
• B.E. Johnson, Cohomology in Banach algebras, American Mathematical Society, Providence, R.I., 1972, Mem. Amer. Math. Soc., No. 127.
• J. Kustermans and S. Vaes, Locally compact quantum groups in the von Neumann algebraic setting, Math. Scand. 92 (2003), no. 1, 68–92.
• V. Losert and H. Rindler, Asymptotically central functions and invariant extensions of Dirac measure, Probability Measures on Groups VII (Herbert Heyer, ed.), Lecture Notes in Math., vol. 1064, Springer Berlin Heidelberg, 1984, 368–378.
• A.L.T. Paterson, Amenability, Math. Surveys and Monogr., vol. 29, Amer. Math. Soc., Providence, RI, 1988.
• Z.-J. Ruan, The operator amenability of $A(G)$, Amer. J. Math. 117 (1995), no. 6, 1449–1474.
• Z.-J. Ruan and G. Xu, Splitting properties of operator bimodules and operator amenability of Kac algebras, Operator theory, operator algebras and related topics (Timişoara, 1996), Theta Found., Bucharest, 1997, 193–216.
• V. Runde, Uniform continuity over locally compact quantum groups, J. Lond. Math. Soc. (2) 80 (2009), no. 1, 55–71.
• R. Stokke, Approximate diagonals and Følner conditions for amenable group and semigroup algebras, Studia Math. 164 (2004), no. 2, 139–159.
• R. Stokke, Quasi-central bounded approximate identities in group algebras of locally compact groups, Illinois J. Math. 48 (2004), no. 1, 151–170.
• B. Willson, Reiter nets for semidirect products of amenable groups and semigroups, Proc. Amer. Math. Soc. 137 (2009), no. 11, 3823–3832.