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On the cohomology of weakly almost periodic group representations. (English) Zbl 1298.22005

In this very nice and well written paper the authors generalize many results on the reduced \(\ell^p\)-cohomology of discrete groups and some results of Y. Shalom [Invent. Math. 141, No. 1, 1–54 (2000; Zbl 0978.22010)] to the setting of weakly almost periodic representations of a group.
Let \(V\) be a Banach space over the real or complex numbers. Let \(B(V)\) denote the algebra of bounded linear operators on \(V\) and \(GL(V)\) the subgroup of \(B(V)\) consisting of the invertible operators on \(V\). Two important topologies that can be placed on \(B(V)\), and of course \(GL(V)\), are the strong operator topology \((sot)\) and the weak operator topology \((wot)\). Let \(G\) be a group. A linear representation \(\rho : G \rightarrow GL(V)\) is weakly almost periodic (wap) if and only if \(\rho(G)\) is relatively \(wot\)-compact in \(B(V)\). Also a Banach \(G\)-module \(V\) is said to be continuous if the corresponding linear representation is \(sot\)-continuous, which means that for every \(v \in V\), the map \(g \in G \mapsto \rho(g)v \in V\) is continuous with respect to the norm on \(V\). Denote by \(V^G\) the subspace of \(V\) of vectors that are invariant under the action of \(G\). The reduced continuous cohomology in degree \(n\) of a topological group \(G\) with coefficients in a Banach \(G\)-module will be denoted by \(\overline{H}^n_c (G, V)\). The main result of this paper is:
Let \(G\) be a topological group and \(V\) be a continuous \(wap\) Banach \(G\)-module. Assume that \(N\) and \(C\) are subgroups of \(G\) with \(C\) lying in the centralizer of \(N\). If \(V^C = \{0\}\), then the restriction map \[ \overline{H}^n_c(G, V) \rightarrow \overline{H}^n_c(N, V) \] in reduced continuous cohomology is zero for every \(n \geq 0\).
Several consequences of this result are also given. One such result is:
Let \(G\) be a topological group and \(Z \leq G\) its center. Let \(V\) be a continuous \(wap\) Banach \(G\)-module such that \(V^Z = \{ 0 \}\). Then for every \(n \geq 0\) \[ \overline{H}^n_c (G, V) = 0. \]
This results generalizes an unpublished result of Elias Kappos concerning the reduced \(\ell^p\)-cohomology of discrete groups. More precisely, if a discrete group has property \(FP_n\) and infinite center, then \(\overline{H}^i (G, \ell^p(G)) = 0\) for \(i \leq n\). Also an example is given that shows that the \(wap\) condition in the hypothesis of the above result cannot be dropped.
The paper also presents several examples of \(wap\) representations of groups.

MSC:

22A25 Representations of general topological groups and semigroups
20J06 Cohomology of groups
43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
22D12 Other representations of locally compact groups

Citations:

Zbl 0978.22010
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References:

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