Amenable Representations and Finite Injective Von Neumann Algebras

Let U(M) be the unitary group of a finite, injective von Neumann algebra M . We observe that any subrepresentation of a group representation into U(M) is amenable in the sense of Bekka; this yields short proofs of two known results—one by Robertson, one by Haagerup—concerning group representations into U(M). A unitary representation π of a group Γ on a Hilbert space Hπ is amenable if there exists on B(Hπ) an Adπ-invariant state, i.e. a state φ on B(Hπ) such that, for any T ∈ B(Hπ), g ∈ Γ: φ(π(g)Tπ(g−1)) = φ(T ). This notion was introduced and studied by Bekka in [Be]. In the present paper, M will always denote a finite, injective von Neumann algebra. We start with the observation that, if π(Γ) is contained in the unitary group U(M), then any subrepresentation of π is amenable. We use this to give short, hopefully new proofs of two known results. The first, due to Robertson ([Ro], Theorem C and remark (4) on p. 554), states that for any representation π of a group Γ with Kazhdan’s property (T) into U(M), the closure of π(Γ) in the L-norm topology on U(M) is compact. The second, due to Haagerup ([Ha], Lemma 2.2), says that for any n ∈ N, U1, U2, . . . , Un ∈ U(M) and P a non-zero projector in the commutant M ′ of M : ∥∥∥∥ n ∑ i=1 PUi ⊗ P̄ Ūi ∥∥∥∥ = n (where the bar denotes the same operator, but acting on the conjugate Hilbert space). Proposition 1. Let π be a representation of a group Γ into U(M). Then any subrepresentation of π is amenable. Proof. We may assume that M = π(Γ)′′. Let ρ be a subrepresentation of π on a closed subspace Hρ which is the range of a projector p ∈ M ′. To construct an Ad ρ-invariant state on B(Hρ) = pB(Hπ)p, choose a conditional expectation Received by the editors October 6, 1995 and, in revised form, December 5, 1995. 1991 Mathematics Subject Classification. Primary 22D25; Secondary 46L10.