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On complements of Kazhdan projections in semisimple groups

We prove that for an isometric representation of some groups on certain Banach spaces, the complement of the subspace of invariant vectors is 1-complemented.


Introduction
An isometric representation π of a locally compact group G on a reflexive Banach space E induces a direct sum decomposition E = E π ⊕ E π into the invariant vectors E π and its canonical, π-invariant complement, E π . The corresponding projection P π from E to E π along E π has norm 1 under fairly general conditions. However, typically, the complementary projection I − P π need not be of norm 1. An easy example is that of the regular representation of a finite group G on ℓ p (G), p > 2, where E π are the constant functions and E π are the functions on G whose mean is zero (see Example 4 for details).
In this article we give conditions under which, for representations π on a Banach space E in a certain class O , consisting of uniformly convex uniformly smooth Banach spaces with the Opial property (see Section 2.1 for details), E π is 1-complemented and more precisely that I − P π = 1. Theorem 1. Let G be a locally compact group and let π be an isometric representation on a Banach space E ∈ O . Assume that there exists a sequence of elements {g n } ⊂ G, g n → ∞, such that Then E π is Birkhoff-James orthogonal to E π and I − P π = 1.
The above theorem in particular applies to class of groups satisfying certain algebraic conditions, introduced as quasi-semisimple groups in [2]. They are defined via the existence of a K AK -type decomposition and generation by certain contraction subgroups, see [2,9] for details. This class of groups includes the classical semisimple Lie groups as well as certain automorphism groups of trees.
Recall that a representation π of G on a reflexive Banach space E is a c 0representation if the matrix coefficients 〈π g v, w〉 vanish at infinity for every v ∈ E π and w ∈ E * . As shown in [2], for quasi-semisimple groups this is the case for all isometric representations on reflexive Banach spaces.

Corollary 2.
Let π be a representation of a non-compact locally compact group G on E ∈ O . If π is a c 0 -representation then 1 − P π = 1.
In particular, if G is quasi-semisimple then the above estimate holds for all isometric representations on E ∈ O . This is particularly interesting in the context of properties such as property (T ℓ p ), studied by Bekka and Olivier [4]. In that case we obtain a stronger conclusion that holds not only in the algebra of bounded operators on E ∈ O , but in the the associated maximal group Banach algebra (see Section 3.1).

Corollary 3.
Let G be quasi-semisimple. Assume that G satisfies uniform property (T O ), namely that there exists a Kazhdan projection p in the Banach algebra C O max (G). Then the element 1 − p satisfies 1 − p = 1.
We remark that in [5] Bernau and Lacey studied projections P such that both P and I − P are of norm 1, and in particular gave a general description of such projections in L p -spaces.

Banach spaces, duality mappings, and the Opial property 2.1 Duality mappings
Consider a duality mapping J : E → S (E * ), into the collection S (E * ) of subsets of E * , defined by the condition A thorough discussion of duality mappings in Banach spaces can be found in e.g. [8].
Denote by w − lim the limit in the weak topology on E. A Banach space E has the Opial property if for every weakly convergent sequence {x n } ⊆ E with the weak limit x 0 ∈ E the inequality lim inf x n − x 0 < lim inf x n − y , holds for every y = x 0 . Every separable Banach space admits an equivalent norm with the Opial property [11].
It is known that a uniformly convex uniformly smooth Banach space has Opial's property if and only if the following condition is satisfied: for every sequence {x n } in E, we have The second mode of convergence is the so-called ∆-convergence. We refer to [18] for details.
By O we will denote the class of uniformly convex uniformly smooth Banach spaces with the Opial property. It is known for instance that the spaces ℓ p , 1 < p < ∞, belong to the class O , as do their infinite direct q-sums, 1 < q < ∞. However, the spaces L p [0, 1], p ∈ (1, 2) ∪ (2, ∞) do not belong to the class O .

Representations
Let B(E) denote the space of bounded linear operators on E. We will consider B(E) with the strong operator topology (SOT) and with the weak operator topology (WOT). Let G be a locally compact group. Let π be an isometric representation of G on E ∈ O that is continuous in the strong operator topology on B(E). In other words, for every v ∈ E the orbit map G → E, g → π g v, is continuous into the norm topology on E. The dual space E * is naturally equipped with an isometric representation π, defined by the formula . When E is reflexive we will say that π is a c 0 -representation if ψ v,w ∈ C 0 (G) for every v ∈ E π and w ∈ E * .

Decompositions and projections
Denote If E is reflexive then there is a direct sum decomposition E = E π ⊕ E π , where E π is the annihilator of the space (E * ) π . Since E π is π-invariant, the decomposition is in fact a decomposition of π into the trivial representation and its complement. See e.g. [1,3]. Note however, that if E is not reflexive then the projection onto E π might not be equivariant [1, Example 2.29], and in fact there might not be a bounded projection onto E π at all [17, Theorem 1]. Let P π : E → E π be the projection along E π . The projection P π is known to satisfy P π = 1 in many cases, see [12,17]. However, for the complementary projection in general we only have In the case p = ∞ and the group Z n , n ≥ 3 it is easy to see that I − P π > 1 by considering the vector f = (1, 1, . . ., 1, −1) ∈ ℓ ∞ (Z n ). Then f = 1, M f = 1−2/n and f − M f = 2−2/n. Since G is finite, clearly, by choosing n ∈ N and 2 < p < ∞, sufficiently large we obtain that the norm of I − P π can be arbitrarily close to 2. The same is true for any group with a finite quotient of cardinality at least 3.
Observe that if G is residually finite with a sequence {N i } of finite index normal subgroups, N i = {e}, then the previous case shows that for the representation of G on ℓ p ( G/N i ) = ( ℓ p (G/N i )) (p) for p = ∞ the norm I − P π will be in fact 2. We can thus choose p such that the norm of the projection I − P π will be arbitrarily close to 2.

Proofs
Before proving the main theorem we first need a few lemmas regarding duality mappings and their behavior with respect to isometric representations within the class O of Banach spaces. For an invertible isometry S on E denote S = (S * ) −1 .
By uniform convexity and uniform smoothness of E the duality mapping J is bijective, which guarantees that there is a unique element w ∈ E * such that w = v and 〈v, w〉 = v 2 . This yields and the claim is proved.
for every λ ∈ R. Given subspaces V 1 , V 2 in E we will say that V 1 is Birkhoff-James orthogonal to V 2 if for every v 1 ∈ V 1 and v 2 ∈ V 2 , v 1 is Birkhoff-James orthogonal to v 2 .
A useful lemma by Kato [13] states that v ≤ v + λw for every λ > 0 if and only if we have 〈w, J(v)〉 ≥ 0.
In particular this implies that v is Birkhoff-James orthogonal to w if we have 〈w, J(v)〉 ≥ 0 and 〈w, J(−v)〉 ≥ 0. Of course, this is equivalent to 〈w, J(v)〉 = 0.
Theorem 1. Let G be a locally compact group and let π be an isometric representation on a Banach space E ∈ O . Assume that there exists a sequence of elements {g n } ⊂ G, g n → ∞, such that Then E π is Birkhoff-James orthogonal to E π and I − P π = 1.
Proof. Let w ∈ E π , v ∈ E π be arbitrary. We first observe that since P π = WOT − lim π g n , then (P π ) * = WOT − limπ * g n = WOT − lim π −1 g n . Thus we have where the last equality follows from Lemma 5. Now, by assumption, the sequence π −1 g n v converges weakly to P π v = 0 since v ∈ E π . By the Opial property we have that this is the same as the condition that J(π −1 g n v) → 0 in the weak topology in E * . In particular, lim w, J(π g n v) = 〈w, J(v)〉 = 0.
The norm estimate I−P π = 1 is then a direct consequence of Birkhoff-James orthogonality.
The above theorem applies to the class of quasi-semisimple groups, studied in [2]. These groups are defined as satisfying two algebraic conditions: a version of a K AK -decomposition and that the group is generated by certain contraction subgroups, associated to sequences of elements tending off to infinity. We refer to [2] and [9] for definitions and applications.
Proof of Corollary 2. If π is an isometric c 0 -representation of G on a reflexive Banach space E then, for v ∈ E π and w ∈ E * , every matrix coefficient of π of the form ψ v,w = 〈π g v, w〉 vanishes as g goes to infinity. This in particular means, that the WOT closure of π(G) in B(E) is the one-point compactification of G, with the point at infinity being precisely the projection P π . In particular, this ensures that the assumptions of Theorem 1 are satisfied.
The class of quasi-semisimple groups includes the classical semisimple Lie groups and certain automorphism groups of trees. As proved in [2], quasi-semisimple groups satisfy a Veech decomposition of their space W(G) of weakly almost periodic functions, (See also [19] for the classical result of Veech on semisimple Lie groups.) Every matrix coefficient of an isometric representation on a reflexive Banach space E is an element of W(G), with the projection (invariant mean on weakly almost periodic functions) m : W(G) → C given by m(ψ v,w ) = 〈P π v, P π w〉. Thus, the previous argument applies to every isometric representation π of a quasisemisimple group on E ∈ O .

Complements of Kazhdan projections
Let O be a class of Banach spaces. The group ring CG can be completed in the the norm f = sup π( f ) , where the supremum is taken over all isometric representations of G on Banach spaces E ∈ O . This completion is a Banach algebra, which we denote C O max (G). In the case when O consists of the class of Hilbert spaces the algebra is the maximal group C * -algebra of G. A Kazhdan projection p ∈ C O max (G) is an idempotent such that π(p) = P π for every isometric representation of G on E ∈ O . G has uniform property (T O ) if a Kazhdan projection exists in p ∈ C O max (G). See [10] for a detailed study.
Proof of Corollary 3. Follows from the fact that for every isometric representation of G on E ∈ O we have I − P π = 1.