##
**Amenability and exactness for dynamical systems and their \(C^*\)-algebras.**
*(English)*
Zbl 1035.46039

The author considers locally compact groups, denoted \(G\). Well-known relations between amenability of \(G\) and nuclearity of its C\(^{*}\)-algebras are extended to amenability of transformation groups \((X,G)\), where \(G\) is acting as transformations of a locally compact \(X\), and also to C*-dynamical systems. For \(G\) acting as automorphisms of a C*-algebra \(A\), the crossed product is well-defined and the corresponding reduced C*-dynamical system is denoted by \(C^{*}_{r}(G,A)\). Of course, \(A\) could be taken to be the commutative C*-algebra of continuous functions on \(X\) vanishing at infinity; for the action \(\alpha :G \to X\), the semi-direct product \(X \times_{\alpha} G\) is \(X \times G\) provided with the product \((x,g).(\alpha_{g^{-1}}(x),h) := (x,gh)\). Operator algebras generated by representations of the semi-direct product are called cross-product algebras [T. Turumaru, Tôhoku Math. J. (2) 10, 355–356 (1958; Zbl 0087.31803)] or covariance algebras [cf. S. Doplicher, D. Kastler and D. W. Robinson, Commun. Math. Phys. 3, 1–28 (1966; Zbl 0152.23803)]. The reviewer feels that, even though the semi-direct product forms a groupoid, it was inapropriate to rely on groupoid theory of the monograph [C. Anantharaman-Delaroche and J. Renault, “Amenable groupoids”, (Monogr. Enseign. Math. 36. Geneva) (2000; Zbl 0960.43003)].

Amenability for \((X,G)\), i.e., an amenable action of \(G\) on \(X\), is taken to be the existence of a approximate invariant continuous mean, loosely speaking a net \((m_{i})\), uniformly convergent on compact subsets of \(X \times G\), of continuous mappings from \(X\) to probability measures on \(G\). This can be defined in several equivalent ways, about the simplest being the existence of a convergent net of continuous positive-definite functions, with compact support in \(X\times G\), which tends to \(1\) uniformly on compact subsets of \(X \times G\) (cf. amenability of \(G\) in the sense that the trivial representation is weakly contained in the regular representation [R. Godement (circa 1950) and P. Urysohn (circa 1925)].

There is no natural inner amenability for transformation groups (cf., for groups, the existence of a mean on \(L^{\infty}(G)\) invariant under inner automorphisms) but the author introduces in its stead a property, denoted (W), expressed in terms of positive-definite functions with compact support in on \((X \times X,G \times G)\); for any compact \(K\) in \(X \times G\) there must exists a ‘properly supported’ continuous positive definite function \(h\) on \((X \times G) \times (X \times G)\) such that \(h(x,t,x,t)\) is close to \(1\) for every \((x,t)\). This condition is weaker than inner amenability in the sense that for groups (i.e., \(X\) reduced to a point) every inner amenable G has property (W). The author proves that \((X,G)\) is amenable if and only it has property (W) and \(C^{*}_{r}(X \rtimes G)\) is nuclear.

The author calls \(G\) amenable at infinity if it admits an amenable action on a compact \(X\) and shows that if \(G\) is countably-locally-compact amenable at infinity it is uniformly embeddable in a Hilbert space [see G. Yu, Invent. Math. 139, 201–240 (2000; Zbl 0956.19004)] and M. E. B. Bekka, P. A. Cherix and A.Vallette [Lond. Math. Soc. Lect. Note Ser. 227, 1–4 (1995; Zbl 0959.43001)].

Exactness of \(G\) (or rather of \(C^{*}_{r}(G)\)) is the preservation of short exact sequences for a C*-algebra \(A\) and two-sided ideal to the corresponding sequence for the \(C^{*}_{r}(G,A)\). If \(G\) is amenable at infinity then \(C^{*}_{r}(G)\) is exact; the author shows also that if G has property (W) and \(C^{*}_{r}(G)\) is exact then \(G\) is amenable at infinity.

There is a discussion on definitions of weak and strong amenability for \((X,G)\); the former a definition for a group and the latter for a transformation group. There is also a proof of a known result on the equivalence, for a discrete group action, of amenability and nuclearity of \(C^{*}(X \rtimes G)\).

Amenability for \((X,G)\), i.e., an amenable action of \(G\) on \(X\), is taken to be the existence of a approximate invariant continuous mean, loosely speaking a net \((m_{i})\), uniformly convergent on compact subsets of \(X \times G\), of continuous mappings from \(X\) to probability measures on \(G\). This can be defined in several equivalent ways, about the simplest being the existence of a convergent net of continuous positive-definite functions, with compact support in \(X\times G\), which tends to \(1\) uniformly on compact subsets of \(X \times G\) (cf. amenability of \(G\) in the sense that the trivial representation is weakly contained in the regular representation [R. Godement (circa 1950) and P. Urysohn (circa 1925)].

There is no natural inner amenability for transformation groups (cf., for groups, the existence of a mean on \(L^{\infty}(G)\) invariant under inner automorphisms) but the author introduces in its stead a property, denoted (W), expressed in terms of positive-definite functions with compact support in on \((X \times X,G \times G)\); for any compact \(K\) in \(X \times G\) there must exists a ‘properly supported’ continuous positive definite function \(h\) on \((X \times G) \times (X \times G)\) such that \(h(x,t,x,t)\) is close to \(1\) for every \((x,t)\). This condition is weaker than inner amenability in the sense that for groups (i.e., \(X\) reduced to a point) every inner amenable G has property (W). The author proves that \((X,G)\) is amenable if and only it has property (W) and \(C^{*}_{r}(X \rtimes G)\) is nuclear.

The author calls \(G\) amenable at infinity if it admits an amenable action on a compact \(X\) and shows that if \(G\) is countably-locally-compact amenable at infinity it is uniformly embeddable in a Hilbert space [see G. Yu, Invent. Math. 139, 201–240 (2000; Zbl 0956.19004)] and M. E. B. Bekka, P. A. Cherix and A.Vallette [Lond. Math. Soc. Lect. Note Ser. 227, 1–4 (1995; Zbl 0959.43001)].

Exactness of \(G\) (or rather of \(C^{*}_{r}(G)\)) is the preservation of short exact sequences for a C*-algebra \(A\) and two-sided ideal to the corresponding sequence for the \(C^{*}_{r}(G,A)\). If \(G\) is amenable at infinity then \(C^{*}_{r}(G)\) is exact; the author shows also that if G has property (W) and \(C^{*}_{r}(G)\) is exact then \(G\) is amenable at infinity.

There is a discussion on definitions of weak and strong amenability for \((X,G)\); the former a definition for a group and the latter for a transformation group. There is also a proof of a known result on the equivalence, for a discrete group action, of amenability and nuclearity of \(C^{*}(X \rtimes G)\).

Reviewer: Aubrey Wulfsohn (Coventry)

### MSC:

46L05 | General theory of \(C^*\)-algebras |

46L55 | Noncommutative dynamical systems |

43A07 | Means on groups, semigroups, etc.; amenable groups |

### Keywords:

C*-algebra; transformation group; semi-direct product; crossed product; amenable; inner amenable; nuclear; groupoid
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\textit{C. Anantharaman-Delaroche}, Trans. Am. Math. Soc. 354, No. 10, 4153--4178 (2002; Zbl 1035.46039)

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