On the invariant uniform Roe algebra
Takeshi Katsura, Otgonbayar Uuye

TL;DR
This paper characterizes when a countable discrete group has the approximation property using the invariant uniform Roe algebra, linking group properties with operator algebraic structures and answering a question posed by Zacharias.
Contribution
It establishes a new equivalence for the approximation property involving the invariant uniform Roe algebra and extends characterizations of group properties to this algebra.
Findings
The approximation property of a group is equivalent to exactness and a tensor product condition involving the invariant uniform Roe algebra.
Characterizations of several group properties extend to the invariant uniform Roe algebra.
Answers a previously open question of J. Zacharias regarding the approximation property.
Abstract
Let be a countable discrete group. We show that has the approximation property if and only if is exact and for any operator space we have , where is the uniform Roe algebra with the right adjoint -action. This answers a question of J. Zacharias. We also show that characterisations of several properties of in terms of the reduced group \cast-algebra apply to the invariant uniform Roe algebra .
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Taxonomy
TopicsAdvanced Operator Algebra Research · Advanced Banach Space Theory · Advanced Topics in Algebra
