Exactness of the reduced crossed product functor

A locally compact group $G$ is called exact (or C*-exact) if for any short exact sequence of C*-algebras

$0\to I\to A \to A/I \to 0$

where each algebra is endowed with a strongly continuous $G$ -action making the sequence equivariant, the associated sequence or reduced crossed products

$0 \to I\rtimes_{r}G\to A\rtimes_{r}G\to (A/I)\rtimes_{ r}G\to 0$

is also exact. I.e. $-\rtimes_{r}G$ is an exact functor from the category of $G$ -C*-algebras (with equivariant morphisms) to the category of C*-algebras.

As opposed to the full crossed product functor, which is always exact in the above sense, there are groups for which the reduced crossed product functor is not exact. For a reference, here are two pathological examples of non-exact groups

Colloquially known as the Gromov monsters, these are defined in [1] and are non-exact discrete groups. There seems to be no imbedding of these groups into some $B(H)$ for any Hilbert space $H$ , so they are unlikely to pop up if one sticks to matrix groups.
Osajda produced a residually finite non-exact groups.

It turns out that if $G$ is any discrete group we have

$C^*_r(G) \text{ is an exact C*-algebra} \quad \Leftrightarrow \quad - \rtimes_{r, \alpha}G \text{ is an exact functor}$

Recall that a C*-algebra $A$ is called exact if $- \otimes_{min}A$ is an exact functor on the category of C*-algebras. Note that the $\Leftarrow$ implication is always true for any locally compact group since if $G$ acts trivial on a C*-algebra $A$ , then $A\otimes_{min}C^r(G) \simeq A\rtimes{\alpha, r}G$ . This statement is proved in [2][Theorem 5.2]. In the same paper the following sufficient conditions are also proved:

Any almost-connected locally compact group, (meaning a locally compact group $G$ for which $G/G^0$ is compact) is exact. In the case of linear algebraic groups, these are characterized by the Borel-Harish-Chandra theorem which states that for a linear algebraic group $G$ over $\Q$ , $G(\R)/\Gamma$ is compact if and only if the $\Q$ -rational morphisms $\phi: G^0\to \C$ are trivial and $G(\Q)$ consists of semisimple components.
Any closed subgroups of an exact groups are exact.
If $N\subset G$ is a normal subgroup with both $N$ and $G/N$ exact, then $G$ is exact.

Furthermore, it is proved in [3] that for any field $K$ and any integer $n>0$ the reduced group C*-algebra of every subgroup of $GL_n(K)$ is exact.

In particular we have the following corollary –

Corollary If $K$ is any field and $\Gamma$ is a discrete linear subgroup of $GL_n(K)$ , the functor $-\rtimes_{r} \Gamma$ is exact.

Bibliography

[1] M. Gromov, \textit{Random walk in random groups}. Geom. Funct. Anal., (1)13, 2003
[2] Kirchberg, Eberhard, and Simon Wassermann. “Exact groups and continuous bundles of C*-algebras.” Mathematische Annalen 315.2 (1999): 169-203.
[3] Guentner, Erik, Nigel Higson, and Shmuel Weinberger. “The Novikov conjecture for linear groups.” Publications mathématiques de l’IHÉS 101.1 (2005): 243-268.

Bibliography

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