# Journal of Operator Theory

Volume 72, Issue 2, Fall 2014  pp. 549-556.

On the invariant uniform Roe algebra

Authors:  Takeshi Katsura (1) and Otgonbayar Uuye (2)
Author institution: (1) Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kouhoku-ku, Yokohama 223-8522, Japan
(2) Department of Mathematics, School of Science, National University of Mongolia, Ulaanbaatar, Mongolia

Summary:  Let $\G$ be a countable discrete group. The invariant uniform Roe algebra of $\G$ is the \cast-subalgebra of its uniform Roe algebra consisting of $\G$-invariant elements. We show that $\G$ has the approximation property if and only if $\G$ is exact and the invariant uniform Roe algebra has a certain slice map property. This answers a question of J. Zacharias. We also show that characterisations of several properties of $\G$ in terms of its reduced group \cast-algebra also apply to its invariant uniform Roe algebra.

DOI: http://dx.doi.org/10.7900/jot.2013aug24.2005
Keywords:  uniform Roe algebra, invariant translation approximation property, approximation property, operator approximation property