ON THE NOTION OF RELATIVE PROPERTY (T) FOR INCLUSIONS OF VON NEUMANN ALGEBRAS by JESSE PETERSON and SORIN POPA*

We prove that the notion of rigidity (or relative property (T)) for inclusions of finite von Neumann algebras defined in [Po1] is equivalent to a weaker property, in which no “continuity constants” are required. The notion of relative property (T) (or rigidity) for inclusions of finite von Neumann algebras with countable decomposable center was introduced in ([P1]) by requiring that one of the following conditions (shown equivalent in [P1]) holds true: (0.1). There exists a normal faithful tracial state τ on N such that: ∀ε > 0, ∃F ′ = F (ε) ⊂ N finite and δ = δ(ε) > 0 such that if H is a Hilbert N -bimodule with a vector ξ ∈ H satisfying the conditions ‖〈·ξ, ξ〉 − τ‖ ≤ δ, ‖〈ξ·, ξ〉 − τ‖ ≤ δ and ‖yξ − ξy‖ ≤ δ, ∀y ∈ F , then ∃ξ0 ∈ H such that ‖ξ0 − ξ‖ ≤ ε and bξ0 = ξ0b, ∀b ∈ B. (0.2). There exists a normal faithful tracial state τ on N such that: ∀ε > 0, ∃F = F (ε) ⊂ N finite and δ = δ(ε) > 0 such that if φ : N → N is a normal, completely positive (abreviated c.p. in the sequel) map with τ ◦φ ≤ τ, φ(1) ≤ 1 and ‖φ(x)−x‖2 ≤ δ, ∀x ∈ F , then ‖φ(b)− b‖2 ≤ ε, ∀b ∈ B, ‖b‖ ≤ 1. (0.3). Condition (0.1) above is satisfied for any normal faithful tracial state τ on N . (0.4). Condition (0.2) above is satisfied for any normal faithful tracial state τ on N . This definition is the operator algebra analogue of the Kazhdan-Margulis relative property (T) for inclusions of groups H ⊂ G ([M], [dHVa]), being formulated in the same spirit Connes and Jones defined the property (T) for single von Neumann algebras in ([CJ]), starting from Kazhdan’s property (T) for groups, by using Hilbert bimodules/c.p. maps (i.e., Connes’ correspondences [C2]). Thus, while in the case H = G the relative property (T) of H ⊂ G amounts to the property (T) of G, in the case B = N and N is a factor the relative property (T) of B ⊂ N in the sense of ([P1]) is equivalent to the property (T) of N in the sense of ([CJ]). But there are in fact two possible ways to define the relative property (T) for inclusions of groups H ⊂ G: one requiring that all representations of G that have an almost *Supported in part by nsf-grant 0100883 Typeset by AMS-TEX 1