##
**Conditional expectations of finite index and properties of modules arising from group actions.**
*(English)*
Zbl 1086.46040

J. Math. Sci., New York 123, No. 4, 4340-4362 (2004); translation from Sovrem. Mat. Prilozh. 1, 177-200 (2003).

In this paper, the author deals with the interplay between properties of an action of a discrete group \(G\) on a compact Hausdorff space \(X\) and algebraic and analytical properties of the module of all continuous functions \(C(X)\) over the algebra of invariant continuous functions \(C^G(X)\). The paper is split into several sections as follows.

Section 1 is a short introduction and description of the goal of the article, namely, to investigate characteristics of actions of (amenable) topological groups on a compact space \(X\) that guarantee the existence of a well-defined induced conditional expectation on the corresponding commutative \(C^*\)-algebra \(C(X)\) and the finiteness of its index. In Section 2, some results concerning conditional expectations on \(C^*\)-algebras that have finite index in the sense of Y. Watatani [“Index for \(C^*\)-subalgebras”. Mem. Am. Math. Soc. 424 (1990; Zbl 0697.46024)] are generalized to the case of conditional expectations \(E\) on \(C^*\)-algebras having finite index in the sense of M. Baillet, Y. Denizeau and J.-F. Havet [Compos. Math. 66, 199–236 (1988; Zbl 0657.46041)], i.e., expectations for which there is a finite real \(K\geq 1\) such that the mapping \((K\cdot E-\text{id})\) is positive.

Section 3 considers conditional expectations arising from actions of topological groups. Let \(X\) be a locally compact Hausdorff space and let \(G\) be a topological group continuously acting on \(X\). The \(C^*\)-subalgebra of \(G\)-invariant continuous functions on \(X\) vanishing at infinity is denoted by \(C^G(X)\) and the stabilizer subgroup of \(x\in X\) by \(G_x\). The action of an amenable group \(G\) on \(X\) generates a conditional expectation \(E_{G,am}\) on \(C_0(X)\) if the formula \(E_{G,am}(f)(x)=m_G(f(g^{-1}x))\) defines a continuous function for any \(f\in C_0(X)\) and for some fixed (left) invariant measure \(m_G\) on \(G\). It is proved that the action of an amenable topological group \(G\) can always be replaced by a suitable action on a completely disconnected group which is the quotient group of \(G\) by its connected component of the identity (Theorem 3.7). On the other hand, the influence of the maximal cardinality of the orbit in \(X\) under the group action on the characteristic constant \(K(E_G)\) of the obtained conditional finite-index expectation on \(C_0(X)\) is described (Theorem 3.8). For an arbitrary topological group \(G\) that acts uniformly continuously on a locally compact space \(X\) such that the cardinality of each orbit is bounded by \(\sharp (Gx)\leq k\), for some fixed \(k\in \mathbb N\), a conditional expectation \(E_G: C_0(X)\longrightarrow E_G(C_0(X))\) is defined by the formula \(E_G(f)(x)=\frac{1}{\sharp(G/G_x)}\cdot\sum_{g_a\in T_x}f(g_ax) \quad (x\in X)\). Also in this case it is shown the influence of the maximal cardinality of the orbit in \(X\) under the group action on the characteristic constant \(K(E_G)\) of the obtained conditional finite-index expectation (Theorem 3.12).

Section 4 deals with conditional finite-index expectations on commutative \(C^*\)-algebras \(C(X)\) arising from actions of discrete groups on compact Hausdorff spaces \(X\). This investigation generalizes the results of Watatani [loc.cit.] on actions of finite groups. Here a detailed analysis is made of actions of amenable discrete groups \(G\) on compact Hausdorff spaces \(X\) for which the orbit of each point is finite. It is proved that if all orbits consists of the same finite number of points then the Hilbert \(C^G(X)\)-module \(\{C(X), E(\langle .,. \rangle) \}\) is finitely generated and projective (Theorem 4.6).

Section 5 is dedicated to expose a recent result of V. V. Seregin [Vest. Mosk. Univ. 58, 44–48 (2003; Zbl 1062.46047)] asserting that when the cardinalities of all orbits are uniformly bounded by some number \(N\), then the Hilbert \(C^G(X)\)-module \(\{C(X), E(\langle .,. \rangle) \}\) is \(C^G(X)\)-reflexive (Theorem 5.2). Section 6 investigates converse statements: from some properties of modules, some properties of actions (Theorems 6.3, 6.4, and 6.7) are deduced. In particular, it is proved in Section 7 that if for a uniformly continuous action, \(C(X)\) is a finitely generated, projective module over \(C^G(X)\), then the cardinality of orbits is finite and constant (Theorem 7.6).

Section 8 is centered on some specific properties of actions of (infinite) discrete groups on compact Hausdorff spaces. Some sufficient Lyapunov-type conditions for the existence of natural conditional expectations are obtained (Theorem 8.7). Mainly Hilbert \(C^*\)-module and topological techniques are used in the paper, which also contains many interesting examples that are very helpful in order to place properly all notions and results presented by the author.

Section 1 is a short introduction and description of the goal of the article, namely, to investigate characteristics of actions of (amenable) topological groups on a compact space \(X\) that guarantee the existence of a well-defined induced conditional expectation on the corresponding commutative \(C^*\)-algebra \(C(X)\) and the finiteness of its index. In Section 2, some results concerning conditional expectations on \(C^*\)-algebras that have finite index in the sense of Y. Watatani [“Index for \(C^*\)-subalgebras”. Mem. Am. Math. Soc. 424 (1990; Zbl 0697.46024)] are generalized to the case of conditional expectations \(E\) on \(C^*\)-algebras having finite index in the sense of M. Baillet, Y. Denizeau and J.-F. Havet [Compos. Math. 66, 199–236 (1988; Zbl 0657.46041)], i.e., expectations for which there is a finite real \(K\geq 1\) such that the mapping \((K\cdot E-\text{id})\) is positive.

Section 3 considers conditional expectations arising from actions of topological groups. Let \(X\) be a locally compact Hausdorff space and let \(G\) be a topological group continuously acting on \(X\). The \(C^*\)-subalgebra of \(G\)-invariant continuous functions on \(X\) vanishing at infinity is denoted by \(C^G(X)\) and the stabilizer subgroup of \(x\in X\) by \(G_x\). The action of an amenable group \(G\) on \(X\) generates a conditional expectation \(E_{G,am}\) on \(C_0(X)\) if the formula \(E_{G,am}(f)(x)=m_G(f(g^{-1}x))\) defines a continuous function for any \(f\in C_0(X)\) and for some fixed (left) invariant measure \(m_G\) on \(G\). It is proved that the action of an amenable topological group \(G\) can always be replaced by a suitable action on a completely disconnected group which is the quotient group of \(G\) by its connected component of the identity (Theorem 3.7). On the other hand, the influence of the maximal cardinality of the orbit in \(X\) under the group action on the characteristic constant \(K(E_G)\) of the obtained conditional finite-index expectation on \(C_0(X)\) is described (Theorem 3.8). For an arbitrary topological group \(G\) that acts uniformly continuously on a locally compact space \(X\) such that the cardinality of each orbit is bounded by \(\sharp (Gx)\leq k\), for some fixed \(k\in \mathbb N\), a conditional expectation \(E_G: C_0(X)\longrightarrow E_G(C_0(X))\) is defined by the formula \(E_G(f)(x)=\frac{1}{\sharp(G/G_x)}\cdot\sum_{g_a\in T_x}f(g_ax) \quad (x\in X)\). Also in this case it is shown the influence of the maximal cardinality of the orbit in \(X\) under the group action on the characteristic constant \(K(E_G)\) of the obtained conditional finite-index expectation (Theorem 3.12).

Section 4 deals with conditional finite-index expectations on commutative \(C^*\)-algebras \(C(X)\) arising from actions of discrete groups on compact Hausdorff spaces \(X\). This investigation generalizes the results of Watatani [loc.cit.] on actions of finite groups. Here a detailed analysis is made of actions of amenable discrete groups \(G\) on compact Hausdorff spaces \(X\) for which the orbit of each point is finite. It is proved that if all orbits consists of the same finite number of points then the Hilbert \(C^G(X)\)-module \(\{C(X), E(\langle .,. \rangle) \}\) is finitely generated and projective (Theorem 4.6).

Section 5 is dedicated to expose a recent result of V. V. Seregin [Vest. Mosk. Univ. 58, 44–48 (2003; Zbl 1062.46047)] asserting that when the cardinalities of all orbits are uniformly bounded by some number \(N\), then the Hilbert \(C^G(X)\)-module \(\{C(X), E(\langle .,. \rangle) \}\) is \(C^G(X)\)-reflexive (Theorem 5.2). Section 6 investigates converse statements: from some properties of modules, some properties of actions (Theorems 6.3, 6.4, and 6.7) are deduced. In particular, it is proved in Section 7 that if for a uniformly continuous action, \(C(X)\) is a finitely generated, projective module over \(C^G(X)\), then the cardinality of orbits is finite and constant (Theorem 7.6).

Section 8 is centered on some specific properties of actions of (infinite) discrete groups on compact Hausdorff spaces. Some sufficient Lyapunov-type conditions for the existence of natural conditional expectations are obtained (Theorem 8.7). Mainly Hilbert \(C^*\)-module and topological techniques are used in the paper, which also contains many interesting examples that are very helpful in order to place properly all notions and results presented by the author.

Reviewer: Salvador Hernández (Castellon)

### MSC:

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

46L08 | \(C^*\)-modules |

46L55 | Noncommutative dynamical systems |