A Hilbert space approach to approximate diagonals for locally compact quantum groups
aa r X i v : . [ m a t h . OA ] N ov A HILBERT SPACE APPROACH TO APPROXIMATEDIAGONALS FOR LOCALLY COMPACT QUANTUM GROUPS
BENJAMIN WILLSON ∗ Abstract.
For a locally compact quantum group G , the quantum group al-gebra L ( G ) is operator amenable if and only if it has an operator boundedapproximate diagonal. It is known that if L ( G ) is operator biflat and has abounded approximate identity then it is operator amenable. In this paper, weconsider nets in L ( G ) which suffice to show these two conditions and combinethem to make an approximate diagonal of the form ω W ′∗ ξ ⊗ η where W is themultiplicative unitary and ξ ⊗ η are simple tensors in L ( G ) ⊗ L ( G ). Indeed, if L ( G ) and L ( ˆ G ) both have a bounded approximate identity and either of thecorresponding nets in L ( G ) satisfies a condition generalizing quasicentralitythen this construction generates an operator bounded approximate diagonal.In the classical group case, this provides a new method for constructing approx-imate diagonals emphasizing the relation between the operator amenability ofthe group algebra L ( G ) and the Fourier algebra A ( G ). Introduction
A locally compact group G is amenable if and only if the group algebra L ( G ) isan amenable Banach algebra. Johnson[7] showed this (and defined amenabilityfor Banach algebras) by using a bounded approximate identity and a net of func-tions tending to left invariance (or tending to invariance under multiplication bypositive norm 1 elements of L ( G )). Ruan [11] proved that the Fourier algebraof G is operator amenable (has an operator approximate diagonal) if and only if G is amenable. Again, this can be shown by combining a bounded approximateidentity for A ( G ) and a net of (quasi-central) functions tending to invarianceunder multiplication in A ( G ).In [8], Kustermans and Vaes formalize the notion of a locally compact quantumgroup as a von Neumann algebra with a co-multiplication and left and right Haarweights. The predual of this von Neumann algebra is a Banach algebra whichis analogous to both L ( G ) and A ( G ) in the group setting. Indeed, there is a‘quantum’ version of the Pontryagin duality theorem which extends the dualitybetween L ( G ) and A ( G ). There are various notions of amenability which gener-alize to the quantum group setting. It is natural to ask whether classical results Date : Received: xxxxxx; Revised: yyyyyy; Accepted: zzzzzz. ∗ Corresponding author.2010
Mathematics Subject Classification.
Primary 43A07; Secondary 20G42, 81R50, 22D35,.
Key words and phrases.
Locally compact quantum group, amenability, operator amenability,approximate diagonal, quasicentral approximate identity. about the equivalence of these extends to quantum groups. In particular, how arethe amenability of a locally compact quantum group G and its dual ˆ G related?In section 2, we provide some background including the definitions of strongamenability and co-amenability. These naturally dual conditions are character-ized by the existence of certain nets in the underlying Hilbert space. These netsare related to a left invariant net and a bounded approximate identity (resp.) in L ( G ). On the dual side, they are related to a b.a.i. and left invariant net in L ( ˆ G ).In [12], Ruan and Xu show that for a Kac algebra, the predual of the vonNeumann algebra is operator amenable if it admits a bounded approximate iden-tity and there is a net in the underlying Hilbert space tending to translationinvariance and quasicentrality. In section 3 of this paper, we extend their resultto preduals of von Neumann algberas on locally compact quantum groups. Thecurrent approach also constructs an operator approximate diagonal directly fromthe nets in L ( G ) rather than using machinery in the second dual of L ( G ). Themain result is that if G is strongly amenable and co-amenable and either of thenets satisfies a quasicentral condition then L ( G ) is operator amenable.In section 4, the analogous construction of an operator bounded approximatediagonal for L ( ˆ G ) is described. The quasicentral conditions of the two netsare not dual to each other. We show a result related to a dual version of thisquasicentrality.Other constructions of quasi-central approximate diagonals have been consid-ered by Stokke in [14]. Nets tending to left invariance have been studied forsemidirect products in [15].The nets used throughout this paper are also related tooperator biflatness of L ( G ). It was conjectured by Aristov, Runde, and Spronk[1]that for a locally compact group, A ( G ) is always operator biflat. This remainsan open question. 2. Background
We rely upon the paper of Kustermans and Vaes [8] to provide further detailson locally compact quantum groups and will herein provide only a brief introduc-tion. We will use the notation from [8] with one significant adjustment. We willemphasize the connection to the classical results by referring to a locally compactquantum group as G = ( M, Γ , φ, ψ ) and using L ∞ ( G ) = M , L ( G ) = M ∗ , and L ( G ) = H φ . However, one should not infer from this notation that there is someset ‘ G ’ on which we define spaces of functions. Definition 2.1. A locally compact quantum group G = ( M, Γ , φ, ψ ) consists ofa von Neumann algebra M (or L ∞ ( G )), a comultiplication Γ, and left ( φ ) andright ( ψ ) Haar weights.The unique (as a Banach space) predual of L ∞ ( G ) has a multiplication givenby the pre-adjoint of Γ. This makes the Banach algebra L ( G ) which will bereferred to as the quantum group algebra of G .The left Haar weight φ is used, via the GNS construction, to create the Hilbertspace L ( G ). Using this Hilbert space, we consider L ∞ ( G ) ֒ → B ( L ( G )). Fur-thermore, every ζ ∈ L ( G ) can generate a ω ζ ∈ L ( G ) via: X ( ω ζ ) = h Xζ , ζ i ( G ) APPROACH TO APPROXIMATE DIAGONALS 3 We will also consider several tensor products of the above spaces: L ( G ) ⊗ L ( G ) is the Hilbert space tensor product, L ∞ ( G ) ⊗ V N L ∞ ( G ) is the von Neumanntensor product (in B ( L ( G ) ⊗ L ( G ))), and L ( G ) ˆ ⊗ L ( G ) is the operator spaceprojective tensor product (see [6] for details) which is the predual of L ∞ ( G ) ⊗ V N L ∞ ( G ).There exists a unitary W in B ( L ( G ) ⊗ L ( G )) (in fact, W is in L ∞ ( G ) ⊗ V N L ∞ ( ˆ G ))such that Γ( x ) = W ∗ (1 ⊗ x ) W for x ∈ L ∞ ( G ). We call W the leftmultiplicative unitary. Similarly there is a right multiplicative unitary V andusing V and L ( G ), we can reconstruct L ∞ ( G ) as the σ -strong closure of { ( ω ⊗ ι )( V ) | ω ∈ B ( L ( G )) ∗ } . The dual LCQG, ˆ G is the σ -strong closure of { ( ω ⊗ ι )( W ) | ω ∈ B ( L ( G )) ∗ } ( Here, { ( ω ⊗ ι ) W denotes the slice map of W with ω inthe first component).The left Haar weight gives rise to a modular conjugation J for L ( G ) andthere is a unique left invariant weight for ˆ G that has a corresponding modularconjugation ˆ J .Along with W and V , we can find similar unitaries in B ( L ( G ) ⊗ L ( G )) whichcorrespond to related LCQGs. In particular, there are ˆ W (the left multiplicativeunitary for the dual LCQG), W o p (for the opposite LCQG – G with opposite co-multiplication), and W ′ (for the commutant LCQG – with von Neumann algebra L ∞ ( G ) ′ ). Proposition 2.2.
Among these operators, the following relations are satisfied: W ∗ = ( ˆ J ⊗ J ) W ( ˆ J ⊗ J )ˆ J J = ν i J ˆ J ˆ W = Σ W ∗ Σ V = ( ˆ J ⊗ ˆ J )Σ W ∗ Σ( ˆ J ⊗ ˆ J )ˆ V = ( J ⊗ J ) W ( J ⊗ J ) W o p = Σ V ∗ Σ = ( ˆ J ⊗ ˆ J ) W ( ˆ J ⊗ ˆ J ) W ′ = ˆ V = ( J ⊗ J ) W ( J ⊗ J ) W W W = W W ˆ W ′ = d W op where ν is some positive number (the scaling constant of the quantum group), Σ denotes the flip map for tensors, and the subscripts on W are the leg notation(eg. W = W ⊗ ∈ B ( L ( G ) ⊗ L ( G ) ⊗ L ( G ) ).Proof. See [8] 2.2, 2.12, 2.15, and 4.1. (cid:3)
Definition 2.3.
A locally compact quantum group is co-amenable if L ∞ ( G ) ∗ (with respect to the two Arens products) is unital. Co-amenability is equivalentto the existence of bounded approximate identity for L ( G ). B´edos and Tuset[2]showed that G is co-amenable if and only if there is a net of norm one vectors B. WILLSON ( η β ) β in L ( G ) such that for each ζ ∈ L ( G ):lim β k W ( η β ⊗ ζ ) − ( η β ⊗ ζ ) k = 0 . (CA)Runde[13] remarked that such a net also satisfies the equivalent condition forthe opposite LCQG G o p :lim β k W o p ( η β ⊗ ζ ) − ( η β ⊗ ζ ) k = 0hence such a net generates a 2-sided bounded approximate identity for L ( G ). Definition 2.4.
A locally compact quantum group is strongly amenable if ˆ G isco-amenable. Equivalently, G is strongly amenable if there is a net of norm onevectors ( ξ α ) α in L ( G ) such that for each ζ ∈ L ( G ):lim α k W ( ζ ⊗ ξ α ) − ( ζ ⊗ ξ α ) k = 0 . (SA) Remark . There is another definition of amenability for locally compact quan-tum groups that depends on the existence of a translation invariant mean in thedual of L ∞ ( G ). If G is strongly amenable, then it is amenable, but the converseis an open question. (In the group case, amenability and strong amenability areequivalent - this is related to the equivalence of Reiter’s P1 and P2 conditions).See [2, 4, 5] for investigations into this question. Remark . Johnson’s main result of [7] was to show that a group is amenableprecisely when the group algebra has a certain homological property which healso termed amenability (for Banach algebras). It is possible to characterizethis in terms of the existence of an approximate diagonal in the tensor product L ( G ) ˆ ⊗ L ( G ). Ruan[11] extended Johnson’s notion of amenability to completelycontractive Banach algebras. In particular, he showed that G is amenable ifand only if there is an operator bounded approximate diagonal for A ( G ). ForLCQGs the appropriate operator space structure on L ( G ) is that resulting fromconsidering the predual of the von Neumann algebra operator space structure of L ∞ ( G ). This leads to the following characterization: Definition 2.7.
The quantum group algebra L ( G ) is operator amenable if itadmits an operator bounded approximate diagonal. That is, if there is a net( x γ ) γ in L ( G ) ˆ ⊗ L ( G ) such that for any a ∈ L ( G ): k a · x γ − x γ · a k L ( G ) ˆ ⊗ L ( G ) → , and; (OBAD1) k Γ ∗ ( x γ ) a − a k L ( G ) → . (OBAD2)Here · denotes the natural bimodule action of L ( G ) on L ( G ) ˆ ⊗ L ( G ) which ismultiplication on the left in the first co-ordinate and multiplication on the rightin the second.3. Operator approximate diagonals from nets in L ( G )This section uses the machinery of quantum groups to discuss operator boundedapproximate diagonals for L ( G ). We construct these diagonals by taking thesimple tensors of elements of nets in L ( G ) satisfying (SA) and (CA) and then ( G ) APPROACH TO APPROXIMATE DIAGONALS 5 apply the multiplicative unitary of the commutant quantum group to the simpletensors. This results in a net in L ( G ) ⊗ L ( G ). The vector states associated tothis net generate a net in L ( G ) ˆ ⊗ L ( G ).We begin this section by showing that the constructed net satisfies (OBAD1)for any locally compact quantum group. For a group G (OBAD2) is automat-ically satisfied so the above mentioned construction provides a novel method ofdescribing a bounded approximate diagonal for L ( G ). This approach relies onlyon the LCQG structure, but does seem to require the equivalence of the multi-plicative operators W and W ′ . This requirement can be weakened. The opera-tors only need to be approximately equivalent when applied to the relevant nets.Continuing this approach, we show that if there are nets in L ( G ) (one whichgenerates a bounded approximate identity, and the other which demonstratesstrong amenability) and W ∗ W ′ applied to either of these nets approximates theidentity then we can construct an operator bounded approximate diagonal.The following lemmas will be useful in the sequal. They are straightforwardmanipulations of the pentagonal rule and some of the other properties listed inproposition (2.2). Lemma 3.1.
For any locally compact quantum group, the following equationsinvolving multiplicative unitaries hold: W W ′∗ = W ′∗ W W ; W W ′∗ = W ′∗ W ′∗ W ; and W ∗ W ∗ = ( ˆ J ⊗ ˆ J ⊗ J ) W W ( ˆ J ⊗ ˆ J ⊗ J ) . Proof.
Rewrite W ′ and W ∗ in terms of W and the modular conjugations andapply the pentagonal rule. W W ′∗ = W (1 ⊗ J ⊗ J ) W ∗ (1 ⊗ J ⊗ J )= W ( ˆ J ⊗ J ⊗ J )( ˆ J ⊗ ⊗ W ∗ (1 ⊗ J ⊗ J )= ( ˆ J ⊗ J ⊗ J ) W ∗ W ∗ ( ˆ J ⊗ ⊗ ⊗ J ⊗ J )= ( ˆ J ⊗ J ⊗ J ) W ∗ W ∗ W ∗ ( ˆ J ⊗ J ⊗ J )= W ′∗ W W . For the second result: W W ′∗ = W ( J ⊗ J ⊗ W ∗ ( J ⊗ J ⊗ W ( J ˆ J ⊗ ⊗ J ⊗ J ⊗ W ∗ ( J ⊗ J ⊗ J ˆ J ⊗ ⊗ W W ( ˆ J ⊗ J ⊗ J ⊗ J ⊗ J ˆ J ⊗ ⊗ W W W ( ˆ J J ⊗ ⊗ W ′∗ W ′∗ W . B. WILLSON
For the third result it is not necessary to apply the pentagonal rule, but onlyto notice that parts of simple tensors commute readily with leg tensors providedthose parts are on different ‘legs’. W ∗ W ∗ = ( ˆ J ⊗ ⊗ J ) W ( ˆ J ⊗ ⊗ J )(1 ⊗ ˆ J ⊗ J ) W (1 ⊗ ˆ J ⊗ J )= ( ˆ J ⊗ ⊗ J ) W ( ˆ J ⊗ ˆ J ⊗ W (1 ⊗ ˆ J ⊗ J )= ( ˆ J ⊗ ˆ J ⊗ J ) W W ( ˆ J ⊗ ˆ J ⊗ J ) (cid:3) The following lemma is an adaptation of [12, 3.14].
Lemma 3.2.
Given a unit vector ξ ∈ L ( G ) , the map θ ξ : L ∞ ( G ) ⊗ V N L ∞ ( G ) → B ( L ( G )) given by θ ξ (Λ) = ( ω ξ ⊗ i )( W ′ Λ W ′∗ ) (for Λ ∈ L ∞ ( G ) ) is weak* continuous, unital, and completely positive . Further-more, θ ξ maps L ∞ ( G ) ¯ ⊗ L ∞ ( G ) into L ∞ ( G ) .Proof. Since W ′ ∈ L ∞ ( G ) ′ ⊗ V N L ∞ ( ˆ G ), the first leg of Λ commutes with W ′ . Forsimple tensors X ⊗ Y ∈ L ∞ ( G ) ⊗ L ∞ ( G ), θ ξ ( X ⊗ Y ) = ( ω J ˆ Jξ ⊗ i )(( X ⊗ Y ))For complete details, see [12, page 205]. (cid:3) We now show that it is possible to combine nets in L ( G ) with properties (CA)and (SA) to create a net in L ( G ) ˆ ⊗ L ( G ) which satisfies the first condition of anapproximate diagonal. The second condition is immediately satisfied if W = W ′ ,but if this is not the case, then some additional assumption must be made. Theorem 3.3.
Let G be a locally compact quantum group. Let ε > . Let ζ ∈ L ( G ) with k ζ k = 1 . Suppose that ξ, η ∈ L ( G ) such that k W ( ζ ⊗ ξ ) − ( ζ ⊗ ξ ) k < ε ; and k ω ζ ∗ ω η − ω η ∗ ω ζ k < ε. Then for Λ ∈ L ∞ ( G ) ⊗ V N L ∞ ( G ) (cid:12)(cid:12) ( ω ζ · ω W ′∗ ( ξ ⊗ η ) − ω W ′∗ ( ξ ⊗ η ) · ω ζ ) (Λ) (cid:12)(cid:12) < ε k Λ k Proof.
Consider (cid:12)(cid:12)(cid:0) ω ζ · ω W ′∗ ( ξ ⊗ η ) − ω W ′∗ ( ξ ⊗ η ) · ω ζ (cid:1) (Λ) (cid:12)(cid:12) . We begin by rewriting in terms of the inner product in L ( G ). ( G ) APPROACH TO APPROXIMATE DIAGONALS 7 (cid:12)(cid:12)(cid:0) ω ζ · ω W ′∗ ( ξ ⊗ η ) − ω W ′∗ ( ξ ⊗ η ) · ω ζ (cid:1) (Λ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:0) ω ζ ⊗ ω W ′∗ ( ξ ⊗ η ) (cid:1) ((Γ ⊗ i ) (Λ)) − (cid:0) ω W ′∗ ( ξ ⊗ η ) ⊗ ω ζ (cid:1) (( i ⊗ Γ)(Λ)) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:0) ω ζ ⊗ ω W ′∗ ( ξ ⊗ η ) (cid:1) ( W ∗ Λ W ) − (cid:0) ω W ′∗ ( ξ ⊗ η ) ⊗ ω ζ (cid:1) ( W ∗ Λ W ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:10) (Λ ) W W ′∗ ( ζ ⊗ ξ ⊗ η ) , W W ′∗ ( ζ ⊗ ξ ⊗ η ) (cid:11) − (cid:10) (Λ ) W W ′∗ ( ξ ⊗ η ⊗ ζ ) , W W ′∗ ( ξ ⊗ η ⊗ ζ ) (cid:11)(cid:12)(cid:12) By using the pentagonal rule results of Lemma (3.1) this becomes: . . . = |h (Λ ) W ′∗ W W ( ζ ⊗ ξ ⊗ η ) , W ′∗ W W ( ζ ⊗ ξ ⊗ η ) i− h (Λ ) W ′∗ W ′∗ W ( ξ ⊗ η ⊗ ζ ) , W ′∗ W ′∗ W ( ξ ⊗ η ⊗ ζ ) i| We now have a W ( ζ ⊗ ξ ) in the first term which is, by assumption, within ε of( ζ ⊗ ξ ). We subtract and add an appropriate term (notice the change in order ofthe tensors) and apply the triangle inequality to get: . . . ≤ |h (Λ ) W ′∗ W W ( ζ ⊗ ξ ⊗ η ) , W ′∗ W W ( ζ ⊗ ξ ⊗ η ) i− h (Λ ) W ′∗ W ( ζ ⊗ ξ ⊗ η ) , W ′∗ W ( ζ ⊗ ξ ⊗ η ) i| + |h (Λ ) W ′∗ W ( ξ ⊗ ζ ⊗ η ) , W ′∗ W ( ξ ⊗ ζ ⊗ η ) i− h (Λ ) W ′∗ W ′∗ W ( ξ ⊗ η ⊗ ζ ) , W ′∗ W ′∗ W ( ξ ⊗ η ⊗ ζ ) i| By the first assumption, the first difference in absolute values is less than 2 ε k Λ k .The final term involves a Λ and a W ′∗ . These commute because their first legsare (respectively) in L ∞ ( G ) and its commutant. So W ′ Λ W ′∗ = Λ and wehave: (cid:12)(cid:12)(cid:0) ω ζ · ω W ′∗ ( ξ ⊗ η ) − ω W ′∗ ( ξ ⊗ η ) · ω ζ (cid:1) (Λ) (cid:12)(cid:12) < ε k Λ k + |h (Λ ) W ′∗ W ( ξ ⊗ ζ ⊗ η ) , W ′∗ W ( ξ ⊗ ζ ⊗ η ) i− h (Λ ) W ′∗ W ( ξ ⊗ η ⊗ ζ ) , W ′∗ W ( ξ ⊗ η ⊗ ζ ) i| = 2 ε k Λ k + |h (1 ⊗ θ ξ (Λ)) W ( ζ ⊗ η ) , W ( ζ ⊗ η ) i− h (1 ⊗ θ ξ (Λ)) W ( η ⊗ ζ ) , W ( η ⊗ ζ ) i| = 2 ε k Λ k + |h (Γ( θ ξ (Λ)))( ζ ⊗ η ) , ( ζ ⊗ η ) i− h (Γ( θ ξ (Λ)))( η ⊗ ζ ) , ( η ⊗ ζ ) i| = 2 ε k Λ k + | ( θ ξ (Λ)) (( ω η ∗ ω ζ )) − ( θ ξ (Λ)) ( ω ζ ∗ ω η )) | < ε k Λ k + ε (cid:13)(cid:13)(cid:13)(cid:16) θ ˆ JJξ (Λ) (cid:17)(cid:13)(cid:13)(cid:13) = 3 ε k Λ k B. WILLSON (cid:3)
Corollary 3.4.
Let G be a strongly amenable and co-amenable locally compactquantum group. Suppose that ( ξ α ) α , ( η β ) β are (SA) and (CA) nets in L ( G ) (respectively). Suppose also that either, for ζ ∈ L ( G ) : k W ∗ ( ξ α ⊗ ζ ) − W ′∗ ( ξ α ⊗ ζ ) k → α
0; (3.1) or k W ∗ ( ζ ⊗ η β ) − W ′∗ ( ζ ⊗ η β ) k → β . (3.2) Then ω W ′∗ ( ξ α ⊗ η β ) has a subnet which is an operator bounded approximate diagonalfor L ( G ) ˆ ⊗ L ( G ) hence L ( G ) is operator amenable.Proof. Runde [13] showed that ω η β is a two sided bounded approximate identityfor L ( G ).For fixed ε >
0, and ζ ∈ L ( G ) there exist (by theorem (3.3)) α and β suchthat (cid:12)(cid:12) ( ω ζ · ω W ′∗ ( ξ α ⊗ η β ) − ω W ′∗ ( ξ α ⊗ η β ) · ω ζ ) (cid:12)(cid:12) < ε for any α (cid:23) α and β (cid:23) β .Now, if (3.1) is true, there is an α (cid:23) α such that for any α (cid:23) α k W ′∗ ξ α ⊗ η β − W ∗ ξ α ⊗ η β k < ε. Since ε and ζ are arbitrary, by choosing α after β there is a subnet of ( ω W ′∗ ξ α ⊗ η β )that is an operator bounded approximate diagonal for L ( G ).If, instead, condition (3.2) is true, then an operator bounded approximatediagonal can be found by choosing β after α . (cid:3) Remark . If ( ξ α ) α is a (SA) net that also satisfies condition (3.1) then itgenerates a bounded approximate identity for L ( ˆ G ), ˆ ω ξ α . This b.a.i. also satisfies k ˆ ω ˆ W op ∗ ˆ W ζ ⊗ ξ α − ˆ ω ζ ⊗ ξ α k → L ( ˆ G ), ˆ ω ζ . This property is related to the notion of quasicentralbounded approximate identities for locally compact groups as studied in [9], [14]and others.4. Dual Version: Operator Amenability of L ( ˆ G )For a locally compact group G , it is well known (eg [6, 10]) that G is amenable ifand only if L ( G ) is (operator) amenable if and only if L ( ˆ G ) = A ( G ) is operatoramenable. It has been conjectured that L ( G ) is operator amenable if and onlyif L ( ˆ G ) is as well. The conditions (CA) and (SA) are naturally dual to eachother, which provides some additional justification for this conjecture. However,recently Caspers, Lee, and Ricard [3] have shown that these two conditions are notsufficient for operator amenability of L ( G ). In this section, we convert the mainresult of section 3 to its natural dual. The dual versions of condition (3.1) and(3.2) interchange the multiplicative unitary for the commutant quantum group( ˆ W ′ ) with that of the quantum group with opposite co-multiplication ( d W op ). Tocreate a more useful dual version, one would hope that if W ′ is approximately ( G ) APPROACH TO APPROXIMATE DIAGONALS 9 W acting on some net in the fashion of (3.1) or (3.2) then W op is approximately W acting on some other net as (4.3) or (4.4). In his proof of the equivalence ofamenability of L ( G ) and operator amenability of A ( G ), Ruan [11] was able touse the fact that, in the group case, W = W ′ to construct a net that workedappropriately with W and W op . Such a nice result does not seem achievable inthe general LCQG case, but we are able to make some progress in this direction.We use an approach motivated in part by a result of Losert and Rindler [9]whereby they construct, for an amenable group, an asymptotically central ap-proximate identity. We are able to combine the two nets of elements of L ( G )(SA net and CA net) to create a bounded approximate identity which also hasan approximately central property as well. Corollary 4.1.
Suppose that ( ξ α ) α , ( η β ) β are nets in L ( G ) such that for every ζ ∈ L ( G ) : k W ( ζ ⊗ ξ α ) − ( ζ ⊗ ξ α ) k →
0; (4.1) k W ( η β ⊗ ζ ) − ( η β ⊗ ζ ) k →
0; (4.2) k W ( ζ ⊗ η β ) − W op ( ζ ⊗ η β ) k → or (4.3) k W ( ξ α ⊗ ζ ) − W op ( ξ α ⊗ ζ ) k → . (4.4) Then ω d W op ∗ ( ξ α ⊗ η β ) is a operator bounded approximate diagonal for L ( ˆ G ) ˆ ⊗ L ( ˆ G ) hence L ( ˆ G ) is operator amenable.Proof. This is a consequence of Corollary (3.4), but rephrased for the dual quan-tum group.If ( ξ α ) α , ( η β ) β are (SA) and (CA) nets for G in L ( G ) (respectively) then theyare also (CA) and (SA) nets for ˆ G (respectively).Furthermore ˆ W ∗ ( ξ ⊗ ζ ) = σ ( W ( ζ ⊗ ξ )) (4.5)and ˆ W ′ ∗ ( ξ ⊗ ζ ) = σ ( W op ( ζ ⊗ ξ )) (4.6)where σ swaps the coordinates in the tensor product.The results follow from corollary (3.4). (cid:3) We use several more lemmas that involve manipulating the multiplicative uni-tary operators.
Lemma 4.2. W ′ op ∗ W ′ W ′ op ∗ W = W ′∗ W ′ op ∗ W ′ W ′ W ′∗ W Proof.
Because W ′ ∈ L ∞ ( G ) ′ ⊗ V N L ∞ ( ˆ G ) and W ′ op ∈ L ∞ ( G ) ′ ⊗ V N L ∞ ( ˆ G ) ′ , itfollows that W ′ and W ′ op ∗ commute.Now, by the pentagonal equation: W ′ W = W ′ W ′ W ′∗ W = W ′∗ W ′ W ′ W ′∗ W and similarly W ′ op ∗ W ′ op ∗ = ˆ J J W ′ W ′ J ˆ J = ˆ J J W ′∗ W ′ W ′ J ˆ J = W ′∗ W ′ op ∗ W ′ . Combining the above two equations, we get the desired result: W ′ op ∗ W ′ W ′ op ∗ W = W ′ op ∗ W ′ op ∗ W ′ W = W ′∗ W ′ op ∗ W ′ W ′∗ W ′ W ′ W ′∗ W = W ′∗ W ′ op ∗ W ′ W ′ W ′∗ W (cid:3) Lemma 4.3. W W W ′ op ∗ = W W ′ op ∗ W W W ′∗ Proof.
By the pentagonal rule we immediately get: W W W ′ op ∗ = W W W W ′ op ∗ By introducing factors of J and ˆ J in the second leg we get: W W ˆ J J W ′∗ W ′ J ˆ J = W W ˆ J J W ′ W ′∗ W ′∗ J ˆ J = W W W ′ op ∗ W W ′∗ Finally, note that the first legs of W and W ′ op ∗ commute, so we get the desiredresult: W W W ′ op ∗ = W W ′ op ∗ W W W ′∗ (cid:3) Theorem 4.4.
Suppose we have nets ( ξ α ) α satisfying conditions (SA) and (3.1) and ( η β ) β satisfying (CA) and (3.2) . For each α, β , consider the element u α,β of L ( G ) given by: u α,β = (1 ⊗ ı ) ω W W op ′∗ ξ α ⊗ η β . Then there exists a subnet u γ = u α γ ,β γ which is a bounded approximate identityfor L ( G ) and satisfies the following quasi-central condition: ( ω ζ ⊗ u γ )( W ′∗ W op ′ Λ W op ′ ∗ W ′ − Λ) → for ζ ∈ L ( G ) and Λ ∈ L ∞ ( G ) ⊗ V N L ∞ ( G ) .Proof. For X ∈ L ∞ ( G ) and ζ ∈ L ( G ), we work towards showing that u γ is abounded approximate identity by considering the value of X ( u α,β ∗ ω ζ ) .By applying lemma (4.3) and noting that X = 1 ⊗ ⊗ X commutes withanything in the first two components, we see that: X ( u α,β ∗ ω ζ )= h X W W W op ′ ∗ ξ α ⊗ η β ⊗ ζ , W W W op ′ ∗ ξ α ⊗ η β ⊗ ζ i = h X W W ′ op ∗ W W W ′∗ ξ α ⊗ η β ⊗ ζ , W W ′ op ∗ W W W ′∗ ξ α ⊗ η β ⊗ ζ i = h X W W W ′∗ ξ α ⊗ η β ⊗ ζ , W W W ′∗ ξ α ⊗ η β ⊗ ζ i . ( G ) APPROACH TO APPROXIMATE DIAGONALS 11 Using this, and the triangle inequality, we see that: | X ( u α,β ∗ ω ζ ) − X ( ω ζ ) | = |h X W W W ′∗ ξ α ⊗ η β ⊗ ζ , W W W ′∗ ξ α ⊗ η β ⊗ ζ i − h Xζ , ζ i|≤ (cid:12)(cid:12) h X W W W ′∗ ξ α ⊗ η β ⊗ ζ , W W W ′∗ ξ α ⊗ η β ⊗ ζ i−h X W W ′∗ ξ α ⊗ η β ⊗ ζ , W W ′∗ ξ α ⊗ η β ⊗ ζ i (cid:12)(cid:12) + (cid:12)(cid:12) h X W W ′∗ ξ α ⊗ η β ⊗ ζ , W W ′∗ ξ α ⊗ η β ⊗ ζ i− h X ξ α ⊗ η β ⊗ ζ , ξ α ⊗ η β ⊗ ζ i|≤ k X k (cid:0) k W W ′∗ ξ α ⊗ ζ − ξ α ⊗ ζ k + k W W ′∗ ξ α ⊗ η β ⊗ ζ − W ′∗ ξ α ⊗ η β ⊗ ζ k (cid:1) Now we consider a fixed Λ ∈ L ∞ ( G ) ⊗ V N L ∞ ( G ) and ζ ∈ L ( G ) to show thequasi-central property. By lemma (4.2) and since Λ , and W ′ commute:( ω ζ ⊗ u γ )( W ′∗ W op ′ Λ W op ′ ∗ W ′ − Λ)= h Λ W op ′ ∗ W ′ W op ′ ∗ W ζ ⊗ ξ α ⊗ η β , W op ′ ∗ W ′ W op ′ ∗ W ζ ⊗ ξ α ⊗ η β i− h Λ W op ′ ∗ W ζ ⊗ ξ α ⊗ η β , W op ′ ∗ W ζ ⊗ ξ α ⊗ η β i = h Λ W ′ ∗ W op ′ ∗ W ′ W ′ W ′ ∗ W ζ ⊗ ξ α ⊗ η β , W ′ ∗ W op ′ ∗ W ′ W ′ W ′ ∗ W ζ ⊗ ξ α ⊗ η β i− h Λ W op ′ ∗ W ζ ⊗ ξ α ⊗ η β , W op ′ ∗ W ζ ⊗ ξ α ⊗ η β i = h Λ W ′ op ∗ W ′ W ′ W ′∗ W ζ ⊗ ξ α ⊗ η β , W ′ op ∗ W ′ W ′ W ′∗ W ζ ⊗ ξ α ⊗ η β i− h Λ W op ′ ∗ W ζ ⊗ ξ α ⊗ η β , W op ′ ∗ W ζ ⊗ ξ α ⊗ η β i So now we have: | ( ω ζ ⊗ u γ )( W ′∗ W op ′ Λ W op ′ ∗ W ′ − Λ) | = |h Λ W ′ op ∗ W ′ W ′ W ′∗ W ζ ⊗ ξ α ⊗ η β , W ′ op ∗ W ′ W ′ W ′∗ W ζ ⊗ ξ α ⊗ η β i− h Λ W op ′ ∗ W ζ ⊗ ξ α ⊗ η β , W op ′ ∗ W ζ ⊗ ξ α ⊗ η β i|≤ |h Λ W ′ op ∗ W ′ W ′ W ′∗ W ζ ⊗ ξ α ⊗ η β , W ′ op ∗ W ′ W ′ W ′∗ W ζ ⊗ ξ α ⊗ η β i− h Λ W ′ op ∗ W ′ W ′ ζ ⊗ ξ α ⊗ η β , W ′ op ∗ W ′ W ′ ζ ⊗ ξ α ⊗ η β i| + |h Λ W op ′ ∗ W ′ W ′ ζ ⊗ ξ α ⊗ η β , W op ′ ∗ W ′ W ′ ζ ⊗ ξ α ⊗ η β i− h Λ W op ′ ∗ W ′ ζ ⊗ ξ α ⊗ η β , W op ′ ∗ W ′ ζ ⊗ ξ α ⊗ η β i| + |h Λ W op ′ ∗ W W ∗ W ′ ζ ⊗ ξ α ⊗ η β , W op ′ ∗ W W ∗ W ′ ζ ⊗ ξ α ⊗ η β i− h Λ W op ′ ∗ W ζ ⊗ ξ α ⊗ η β , W op ′ ∗ W ζ ⊗ ξ α ⊗ η β i|≤ k Λ k ∞ (cid:0) k W ′∗ W ζ ⊗ ξ α ⊗ η β − ζ ⊗ ξ α ⊗ η β k + k W ′ ζ ⊗ ξ α ⊗ η β − ζ ⊗ ξ α ⊗ η β k + k W ∗ W ′ ζ ⊗ ξ α ⊗ η β − ζ ⊗ ξ α ⊗ η β k )For fixed ε >
0, there exists an α ε such that, by (SA) k W ′ ζ ⊗ ξ α ⊗ η β − ζ ⊗ ξ α ⊗ η β k < ε and by (3.1) k W W ′∗ ξ α ⊗ ζ − ξ α ⊗ ζ k < ε Furthmore, there is a β ε such that, by (CA) k W W ′∗ ξ α ⊗ η β ⊗ ζ − W ′∗ ξ α ⊗ η β ⊗ ζ k < ε and by (3.2) k W ′∗ W ζ ⊗ ξ α ⊗ η β − ζ ⊗ ξ α ⊗ η β k < ε and k W ∗ W ′ ζ ⊗ ξ α ⊗ η β − ζ ⊗ ξ α ⊗ η β k < ε So, by choosing β after α there is a subnet u γ which is a bounded approximateidentity satisfying condition (4.7). (cid:3) Converting the result above for L ( G ) into the corresponding result for L ( G )that is desired for the previous result may be difficult. It is worth noting thesimilarity between this challenge and the open problem of whether the existenceof a left invariant mean implies strong amenability. Runde and Daws conjecturedthat some version of Leptin’s theorem would be helpful in the latter case. Itwould perhaps be similarly helpful for the former. Acknowledgement.
The author gratefully acknowledges the financial supportof Hanyang University via a research fund for new professors.The author would like to thank Professor Zhiguo Hu and others at the Univer-sity of Windsor for many helpful comments and guidance in this research.The author would like to thank the referee for his/her careful reading of thepaper and helpful comments and corrections and Yemon Choi for directing theauthor to the paper of Ruan and Xu [12].
References
1. O. Y. Aristov, V. Runde, and N. Spronk,
Operator biflatness of the Fourier algebra andapproximate indicators for subgroups , J. Funct. Anal. (2004), no. 2, 367–387.2. E. B´edos and L. Tuset,
Amenability and co-amenability for locally compact quantum groups ,Internat. J. Math. (2003), no. 8, 865–884.3. M. Caspers, H. H. Lee, and ´E. Ricard, Operator biflatness of the L -algebras of compactquantum groups , J. Reine Angew. Math. (2013).4. M. Daws and V. Runde, Reiter’s properties ( P ) and ( P ) for locally compact quantumgroups , J. Math. Anal. Appl. (2010), no. 2, 352 – 365.5. P. Desmedt, J. Quaegebeur, S. Vaes, et al., Amenability and the bicrossed product construc-tion , Illinois J. Math. (2002), no. 4, 1259–1277.6. E. G. Effros and Z.-J. Ruan, Operator spaces , London Math. Soc. Monogr. New Series,vol. 23, The Clarendon Press, Oxford University Press, New York, 2000.7. B. E. Johnson,
Cohomology in Banach algebras , American Mathematical Society, Provi-dence, R.I., 1972, Mem. Amer. Math. Soc., No. 127.8. J. Kustermans and S. Vaes,
Locally compact quantum groups in the von Neumann algebraicsetting , Math. Scand. (2003), no. 1, 68–92.9. V. Losert and H. Rindler, Asymptotically central functions and invariant extensions ofDirac measure , Probability Measures on Groups VII (Herbert Heyer, ed.), Lecture Notesin Math., vol. 1064, Springer Berlin Heidelberg, 1984, pp. 368–378.10. A. L. T. Paterson,
Amenability , Math. Surveys and Monogr., vol. 29, Amer. Math. Soc.,Providence, RI, 1988. ( G ) APPROACH TO APPROXIMATE DIAGONALS 13
11. Z.-J. Ruan,
The operator amenability of A ( G ), Amer. J. Math. (1995), no. 6, 1449–1474.12. Z.-J. Ruan and G. Xu, Splitting properties of operator bimodules and operator amenabilityof Kac algebras , Operator theory, operator algebras and related topics (Timi¸soara, 1996),Theta Found., Bucharest, 1997, pp. 193–216.13. V. Runde,
Uniform continuity over locally compact quantum groups , J. Lond. Math. Soc.(2) (2009), no. 1, 55–71.14. R. Stokke, Approximate diagonals and Følner conditions for amenable group and semigroupalgebras , Studia Math. (2004), no. 2, 139–159.15. B. Willson,
Reiter nets for semidirect products of amenable groups and semigroups , Proc.Amer. Math. Soc. (2009), no. 11, 3823–3832. Department of Mathematics, School of Natural Sciences, Hanyang Univer-sity, 222 Wangsimni-ro, Seongdong-gu, Seoul 133-791, Korea.
E-mail address ::