# 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

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