Invariant mean on weakly almost periodic functionals over a Hopf-von Neumann algebra
Abstract
Let M be a Hopf--von Neumann algebra with predual M*. Using minimal weak*-closed left ideals, we obtain an idempotent state φ∈WAP(M*)* satisfying φ⋆ ω⋆φ=φ for every state ω on WAP(M*). This state determines the largest closed introverted subspace Xφ⊆WAP(M*) containing the identity and admitting a two-sided invariant mean. We also characterize the existence of a unique two-sided invariant mean on introverted operator systems in WAP(M*) by a symmetry condition on their left and right state orbits. The criterion recovers the classical results for locally compact groups and their Fourier algebras, and gives a unified treatment of cocommutative Hopf--von Neumann algebras, compact quantum groups, unital C*-Eberlein algebras, and WAP(M(G)) for every locally compact group G.
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BibTeX
@misc{faal2026invariant,
title = {{Invariant mean on weakly almost periodic functionals over a Hopf-von Neumann algebra}},
author = {Ramin Faal and Hamid Reza Ebrahimi Vishki and Amin Allah Khosravi},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00431},
eprint = {2611.00431},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00431},
}
RIS
TY - UNPB
TI - Invariant mean on weakly almost periodic functionals over a Hopf-von Neumann algebra
AU - Ramin Faal
AU - Hamid Reza Ebrahimi Vishki
AU - Amin Allah Khosravi
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00431
UR - https://arxiv.org/abs/2611.00431
AB - Let M be a Hopf--von Neumann algebra with predual M_*. Using minimal weak*-closed left ideals, we obtain an idempotent state φ∈WAP(M_*)^* satisfying φ⋆ ω⋆φ=φ for every state ω on WAP(M_*). This state determines the largest closed introverted subspace X_φ⊆WAP(M_*) containing the identity and admitting a two-sided invariant mean. We also characterize the existence of a unique two-sided invariant mean on introverted operator systems in WAP(M_*) by a symmetry condition on their left and right state orbits. The criterion recovers the classical results for locally compact groups and their Fourier algebras, and gives a unified treatment of cocommutative Hopf--von Neumann algebras, compact quantum groups, unital C^*-Eberlein algebras, and WAP(M(G)) for every locally compact group G.
ER -