JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 23, Number 4, October 2010, Pages 1175–1185 S 0894-0347(10)00668-5 Article electronically published on March 26, 2010 B(p) IS NEVER AMENABLE VOLKER RUNDE Introduction The theory of amenable Banach algebras begins with B. E. Johnson’s memoir [Joh 1]. The choice of terminology comes from [Joh 1, Theorem 2.5]: a locally compact group is amenable (in the usual sense) if and only if its L1-algebra satisfies a certain cohomological triviality condition, which is then used to define the class of amenable Banach algebras. The notion of Banach algebraic amenability has turned out to be extremly fruit- ful ever since the publication of [Joh 1]. The class of amenable Banach algebras is small enough to allow for strong theorems of a general nature, yet large enough to encompass a diverse collection of examples, such as the nuclear C∗-algebras (see [Run 2, Chapter 6] for an account), Banach algebras of compact operators on certain well-behaved Banach spaces ([G–J–W]), and even radical Banach algebras ([Run 1] and [Rea 1]). The Memoir [Joh 1] concludes with a list of suggestions for further research. One of them ([Joh 1, 10.4]) asks: Is B(E)—the Banach algebra of all bounded linear operators on a Banach space E—ever amenable for any infinite-dimensional E? From a philosophical point of view, the answer to this question ought to be a clear “no”. As for groups, amenability for Banach algebras can be viewed as a weak finiteness condition: amenable Banach algebras tend to be “small”—whatever that may mean precisely—and B(E) simply feels too “large” to be amenable. It seems, however, as if Johnson’s question has recently—somewhat surprisingly— found a positive answer: in [A–H], S. A. Argyros and R. G. Haydon construct an infinite-dimensional Banach space E with few bounded linear operators, i.e., B(E)=K(E)+C idE (with K(E) denoting the compact operators on E). As H. G. Dales pointed out to the author, E has property (A) introduced in [G–J–W], so that K(E) is an amenable Banach algebra for this space, as is, consequently, B(E). Still, infinite-dimensional Banach spaces E with B(E) amenable ought to be the exception rather than the rule. Indeed, it follows from work by S. Wassermann ([Was]) and the equivalence of amenability and nuclearity for C∗-algebras that B(2) cannot be amenable. With 2 being the “best behaved” of all p-spaces, one is led to expect that B(p) fails to be amenable for all p ∈ [1, ∞]. However, until recently Received by the editors July 4, 2009 and, in revised form, December 5, 2009, December 7, 2009, and December 8, 2009. 2010 Mathematics Subject Classification. Primary 47L10; Secondary 46B07, 46B45, 46H20. Key words and phrases. Amenability, Kazhdan’s property (T ), Lp-spaces. The author’s research was supported by NSERC. c 2010 American Mathematical Society Reverts to public domain 28 years from publication 1175 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1176 VOLKER RUNDE it was not known for any p ∈ [1, ∞] other than 2 whether or not B(p)isamenable. The first to establish the non-amenability of B(p) for any p ∈ [1, ∞] \{2} was C. J. Read in [Rea 2], where he showed that B(1) is not amenable. Subsequently, Read’s proof was simplified by G. Pisier ([Pis]). Eventually, N. Ozawa simplified Pisier’s argument even further and gave a proof that simultaneously establishes the non-amenability of B(p)forp =1, 2, ∞ ([Oza]). In [D–R], M. Daws and the author investigated the question of whether B(p)is amenable for p ∈ (1, ∞) \{2}. Among the results obtained in [D–R] are the follow- ing: If B(p) is amenable, then so are ∞(B(p)) and ∞(K(p)). The amenability of ∞(K(p)), in turn, forces ∞(K(E)) to be amenable for every infinite-dimensional Lp-space in the sense of [L–P]; in particular, if B(p)isamenable,thensois ∞(K(2 ⊕ p)). This last implication is the starting point of this paper. Through a modification of Ozawa’s approach from [Oza], we show that ∞(K(2 ⊕ E)) is not amenable for certain Banach spaces E, including E = p for all p ∈ (1, ∞). As a consequence, B(p) cannot be amenable for such p (and neither can B(Lp[0, 1])). Together with the results from [Rea 2] and [Oza], this proves that B(p) is not amenable for any p ∈ [1, ∞]. 1. Amenable Banach algebras The original definition of an amenable Banach algebra from [Joh 1] is given in terms of first-order Hochschild cohomology. An equivalent, but more intrin- sic characterization—through approximate and virtual diagonals—was given soon thereafter in [Joh 2]. For the work done in this paper, however, yet another equiv- alent characterization of amenability due to A. Ya. Helemski˘ı turns out to be best suited ([Hel, Theorem VII.2.20]). We denote the algebraic tensor product by ⊗ and use the symbol ⊗ˆ for the projective tensor product of Banach spaces. If A and B are Banach algebras, then so is A⊗ˆ B in a canonical fashion. For any Banach algebra A,weuseAop for its opposite algebra, i.e., the underlying Banach space is the same as for A, but multiplication has been reversed. Multiplication in A induces a bounded linear map Δ: A⊗ˆ A → A;itisimmediatethatkerΔisaleftidealinA⊗ˆ Aop. Definition 1.1. A Banach algebra A is said to be amenable if (a) A has a bounded approximate identity and (b) the left ideal ker Δ of A⊗ˆ Aop has a bounded right approximate identity. Definition 1.1 makes the proof of the following lemma, which we will require later on, particularly easy: Lemma 1.2. Let A be an amenable Banach algebra, and let e ∈ A be an idempotent. Then, for any >0 and any finite subset F of eAe, there are a1,b1,...,ar,br ∈ A such that r (1) akbk = e k=1 and r ⊗ − ⊗ ∈ (2) xak bk ak bkx < (x F ). k=1 A⊗ˆ A License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use B(p) IS NEVER AMENABLE 1177 Proof. Let • denote the product in A⊗ˆ Aop.SinceF ⊂ eAe,wehavethat{x ⊗ e − e ⊗ x : x ∈ F }⊂ker Δ. By Definition 1.1(b), there is r ∈ ker Δ such that (3) (x ⊗ e − e ⊗ x) • r − (x ⊗ e − e ⊗ x) <. Set d := e ⊗ e − (e ⊗ e) • r. Without loss of generality, we can suppose that ∈ ⊗ ∈ r ⊗ d A A, i.e., there are a1,b1,...,ar,br A with d = j=1 aj bj .Fromthe definition of d, it is then immediate that (1) holds while (3) translates into (2). Remark. Since ker Δ has bounded right approximate identity, there is C ≥ 0, de- ∈ pending only on A, but not on F or, such that a1,b1,...,ar,br A satisfying (1) r ⊗ ≤ and(2)canbechosensuchthat k=1 ak bk A⊗ˆ A C. We shall make no use of this, however. 2. Ozawa’s proof revisited In [Oza], Ozawa presents a proof that simultaneously establishes the non-amen- B p ∞ ∞ ∞ B p ability of the Banach algebras ( )forp =1, 2, and - n=1 (N ) for all p ∈ [1, ∞]. In this section, we recast the final step of his proof as a lemma, which does not make any reference to particular Banach algebras. A pivotal rˆole in Ozawa’s argument is played by the fact that the group SL(3, Z) has Kazhdan’s property (T ) ([B–dlH–V, Theorem 4.2.5]) and thus, in particular, is finitely generated ([B–dlH–V, Theorem 1.3.1]) by g1,...,gm, say. For the definition (and more) on Kazhdan’s property (T ), we refer to [B–dlH–V]. What we require is the following consequence of SL(3, Z)having(T )([Oza, Theorem 3.1]; compare also [B–dlH–V, Proposition 1.1.9]): Proposition 2.1. For any g1,...,gm generating SL(3, Z), there is a constant κ>0 such that, for any unitary representation π of SL(3, Z) on a Hilbert space H and any ξ ∈ H,thereisaπ-invariant vector η ∈ H, i.e., satisfying π(g)η = η for all g ∈ SL(3, Z), such that ξ − η≤κ max π(gj)ξ − ξ. j=1,...,m We briefly recall the setup laid out in [Oza, Section 3], which we will require both for the lemma at the end of this section and in the proof of Theorem 3.2 below. Let P denote the set of all prime numbers, and let, for each p ∈ P, the projective plane over the finite field Z/pZ be denoted by Λp. Obviously, SL(3, Z)actson Λp through matrix multiplication, which, in turn, induces a unitary representation 2 πp of SL(3, Z)on (Λp). The action of SL(3, Z)onΛp is 2-transitive, i.e., the product action of SL(3, Z)onΛp × Λp has exactly two orbits: the diagonal and its 2 complement. Consequently, whenever η ∈ (Λp × Λp)isπp ⊗ πp-invariant, there are α, β ∈ C such that (4) η = αζ1 + βζ2 with − 1 − | | 2 ⊗ | | 1 ⊗ (5) ζ1 = Λp eλ eλ and ζ2 = Λp eλ eμ. λ∈Λp λ,μ∈Λp (Here, eλ for λ ∈ Λp is the point mass at λ, as usual.) Finally, choose a subset | |− | | Λp 1 ∈B 2 Sp of Λp such that Sp = 2 , and define an invertible isometry vp ( (Λp)) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1178 VOLKER RUNDE through eλ,λ∈ Sp, vpeλ = −eλ,λ/∈ Sp.