On the invariant uniform Roe algebra

2012/11/07 by Takeshi Katsura, Katsura, Takeshi, Otgonbayar Uuye +1
Mathematics · #20F65 #47L30 #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.1211.1501

openalex publication_date 2012/11/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Γ be a countable discrete group. We show that Γ has the approximation property if and only if Γ is exact and for any operator space S ⊆ \K(H) we have \Cu(Γ)^Γ ⊗ S = (\Cu(Γ) ⊗ S)^Γ, where \Cu(Γ) is the uniform Roe algebra with the right adjoint Γ-action. This answers a question of J. Zacharias. We also show that characterisations of several properties of Γ in terms of the reduced group \cast-algebra apply to the invariant uniform Roe algebra \Cu(Γ)^Γ.

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