Amenability constants for semilattice algebras
aa r X i v : . [ m a t h . F A ] A ug AMENABILITY CONSTANTS FOR SEMILATTICEALGEBRAS
MAHYA GHANDEHARI, HAMED HATAMI AND NICO SPRONK
Abstract.
For any finite commutative idempotent semigroup S , a semi-lattice , we show how to compute the amenability constant of its semi-group algebra ℓ ( S ), which is always of the form 4 n + 1. We then showthat these give lower bounds to amenability constants of certain Banachalgebras graded over semilattices. We also demonstrate an example ofa commutative Clifford semigroup G for which amenability constant of ℓ ( G ) is not of the form 4 n + 1. We also show there is no commutativesemigroup with amenability constant between 5 and 9. In conjunction with V. Runde [13], the third named author proved thatfor a locally compact group G , G is compact if and only if its Fourier-Stieltjesalgebra B( G ) is operator amenable with operator amenability constant lessthan 5. In a subsequent article [14], examples of non-compact groups G were found for which the operator amenability constant is exactly 5. In re-lated work of Dales, Lau and Strauss [3, Corollary 10.26], improving on [16,Theorem 3.2], it was shown that a semigroup algebra ℓ ( S ) has amenabilityconstant less than 5, if and only if S is an amenable group. For the multi-plicative semigroup L = { , } , it is known that the amenability constantof ℓ ( L ) is 5. These parallel facts are not coincidences since for the spe-cial groups G , mentioned above, B( G ) is ℓ -graded over L , i.e. there are1-operator amenable subalgebras A and A such that B( G ) = A ⊕ ℓ A ,and A is an ideal.We are thus led to consider the general situation of Banach algebrasgraded over semilattices, i.e. commutative idempotent semigroups, whichwe define in Section 2. To do this, in Section 1 we develop a method forcomputing the amenability constants associated to finite semilattice alge-bras. The results in Section 1 have a similar flavour to some results fromthose in the recent monograph [3], and are very similar to some results ofDuncan and Namioka [4]. However, our method is explicit and quantitative,and thus is a nice complement to their work. In Section 2 we obtain a lowerbound for the amenability constant of Banach algebras graded over finite Mathematics Subject Classification.
Primary 46H20, 43A20; Secondary 20M14,43A30.
Key words and phrases. amenable/contractible Banach algebra, semilattice,graded Banach algebra.Research of the third named author supported by NSERC Grant 312515-05. semilattices. We show a surprising example which indicates our lower boundis not, in general the amenability constant. We show, at least for certainfinite dimensional algebras graded over linear semilattices, that our lowerbound is achieved. We close with an answer to a question asked of us byH.G. Dales: we show that there does not exist a commutative semigroup G such that 5 < AM( ℓ ( G )) < G ). Our hope is that the operator amenability constants AM op (B( G ))can all be computed. We conjecture they are a subset of { n + 1 : n ∈ N } ,motived by Theorem 1.7 and Theorem 2.2, below. We hope that these val-ues will serve as a tool for classifying for which groups G , B( G ) is operatoramenable.Interest in amenability of semigroup algebras, in particular for inversesemigroups and Clifford semigroups, goes back at least as far as Duncanand Namioka [4]. Grønbæk [5] characterised commutative semigroups G for which ℓ ( G ) is amenable. A recent extensive treatise on ℓ -algebras ofsemigroups has been written by Dales, Lau and Strauss [3], which includes acharaterisation of all semigroups G for which ℓ ( G ) is amenable. Biflatnessof ℓ ( S ), for a semilattice S , has recently been characterised by Choi [1].0.1. Preliminaries.
Let A be a Banach algebra. Let A ⊗ γ A denote theprojective tensor product. We let m : A⊗ γ A → A denote the multiplicationmap and we have left and right module actions of A on A ⊗ γ A given onelementary tensors by a · ( b ⊗ c ) = ( ab ) ⊗ c and ( b ⊗ c ) · a = b ⊗ ( ca ) . A bounded approximate diagonal (b.a.d.) is a bounded net ( D α ) in A ⊗ γ A such that ( m ( D α )) is a bounded approximate identity in A , i.e.(0.1) lim α am ( D α ) = a and lim α m ( D α ) a = a for each a in A and ( D α ) is asymptotically central for the A -actions, i.e.(0.2) lim α ( a · D α − D α · a ) = 0 for each a in A . Following Johnson [10], we will say that a Banach algebra A is amenable ifit admits a b.a.d. A quantitative feature of amenability was introduced byJohnson in [11], for applications to Fourier algebras of finite groups. The amenability constant of an amenable Banach algebra A is given byAM( A ) = inf (cid:26) sup α k D α k γ : ( D α ) is a b.a.d. for A (cid:27) . The problem of understanding amenable semigroup algebras in terms oftheir amenability constants has attracted some attention [16, 3].
MENABILITY CONSTANTS 3
We call A contractible if it admits a diagonal , i.e. an element D in A ⊗ γ A for which am ( D ) = a = m ( D ) a and(0.3) a · D = D · a (0.4)for each a in A . Note, in particular, then A must be unital and the normof the unit is bounded above by AM( A ).If A is a finite dimensional amenable Banach algebra, then A ⊗ γ A isa finite dimensional Banach space, so any b.a.d. admits a cluster point D .Since any subnet of a b.a.d. is also a b.a.d., the cluster point must be adiagonal, whence A is contractible.We record the following simple observation. Proposition 0.1. If A is a contractible commutative Banach algebra, thenthe diagonal is unique. Proof.
We note that
A ⊗ γ A is a Banach algebra in an obvious way: ( a ⊗ b )( c ⊗ d ) = ( ac ) ⊗ ( bd ). If D is a diagonal, then ( a ⊗ b ) D = a · D · b = ( ab ) · D for a, b in A , by commutativity. Hence if D ′ is another diagonal D ′ D = m ( D ′ ) · D = 1 · D = D and, similarly, D ′ D = DD ′ = D ′ . (cid:3) It will also be useful to observe the following.
Proposition 0.2.
Let A and B be contractible Banach algebras, with re-spective diagonals D A and D B , then A ⊗ γ B has diagonal D A ⊗ D B ∈ ( A ⊗ γ A ) ⊗ γ ( B ⊗ γ B ) ∼ = ( A ⊗ γ B ) ⊗ γ ( A ⊗ γ B ) . Proof.
It is simple to check the diagonal axioms (0.3) and (0.4). (cid:3) Amenability constants for semilattice algebras A semilattice is a commutative semigroup S in which each element isidempotent, i.e. if s ∈ S then ss = s . If s, t ∈ S we write(1.1) s ≤ t ⇔ st = s. It is clear that this defines a partial order on S . We note that if S is a finitesemilattice, then o = Q s ∈ S s is a minimal element for S with respect to thispartial order. We note that if S has a minimal element, then it is unique.Also if S has a unit 1, then 1 is the maximal element in S .A basic example of a semilattice is P ( T ), the set of all subsets of a set T , where we define στ = σ ∩ τ for σ, τ in P ( T ). The minimal element is ∅ ,and the maximal element is T . We call any subsemilattice of a semilattice P ( T ) a subset semilattice . This type of semilattice is universal as we havea semilattice “Cayley Theorem”: for any semilattice S , the map s
7→ { t ∈ S : t ≤ s } : S → P ( S ) (or s
7→ { t ∈ S \ { o } : t ≤ s } : S → P ( S \ { o } )) is aninjective semilattice homomorphism (by which o ∅ ). MAHYA GHANDEHARI, HAMED HATAMI AND NICO SPRONK
For any semilattice S we define ℓ ( S ) = ( x = X s ∈ S x ( s ) δ s : each x ( s ) ∈ C and k x k = X s ∈ S | x ( s ) | < ∞ ) where each δ s is the usual “point mass” function. Then ℓ ( S ) is a commu-tative Banach algebra under the norm k·k with the product X s ∈ S x ( s ) δ s ! ∗ X t ∈ S x ( t ) δ t ! = X r ∈ S X st = r x ( s ) y ( t ) ! δ r . In particular we have δ s ∗ δ t = δ st . We shall consider the Banach space ℓ ∞ ( S ), of bounded functions from S to C with supremum norm, to be analgebra under usual pointwise operations. The Cayley map, indicated above,extends to an algebra homomorphism Σ : ℓ ( S ) → ℓ ∞ ( S ), given on each δ s by(1.2) Σ( δ s ) = χ { t ∈ S : t ≤ s } and extended linearly and continuously to all of ℓ ( S ). Here, χ T is theindicator function of T ⊂ S . The map Σ is called the Sch¨utzenburger map ;see [1, §
4] and references therein.We note that if S is finite, then Σ is a bijection. In this case a formulafor its inverse is given by(1.3) Σ − ( χ s ) = X t ≤ s µ ( t, s ) δ t where χ s = χ { s } and µ : { ( t, s ) : S × S : t ≤ s } → R is the M¨obius function of the partially ordered set ( S, ≤ ) as defined in [15, § µ , though we willnever need to know µ directly.It follows from [4, Theorem 10] that ℓ ( S ) is amenable if and only if S isfinite. Thus it follows (0.3) that ℓ ( S ) is unital if S is finite. If S is unital,then δ is the unit for ℓ ( S ). If S is not unital, the unit is more complicated.We let M ( S ) denote the set of maximal elements in S with respect to thepartial ordering (1.1). Proposition 1.1. If S is a finite semilattice then the unit is given by u = P p ∈ S u ( p ) δ p where (1.4) u ( p ) = 1 − X t>p u ( t ) for each p in S and we adopt the convention that an empty sum is . More-over (1.5) X s ∈ S u ( s ) = 1 . MENABILITY CONSTANTS 5
Proof.
While we have already established existence of the unit above, letus note that we can gain a very elementary proof of its existence. Indeedsince Σ : ℓ ( S ) → ℓ ∞ ( S ) is a bijection, u = Σ − ( χ S ) is the unit for ℓ ( S ).If p ∈ S then δ p = δ p ∗ u = X s ≥ p u ( s ) δ p + X s
Let S be a finite semilattice. Then the diagonal D = X ( s,t ) ∈ S × S d ( s, t ) δ s ⊗ δ t satisfies, for all ( p, q ) in S × S , (a) d ( p, p ) = u ( p ) − X ( s,t ) > ( p,p ) st = p d ( s, t ) ; (b) if q p , then d ( p, q ) = − X t>q d ( p, t ) and d ( q, p ) = − X s>q d ( s, p ) ; and (c) d ( p, q ) = d ( q, p ) .Thus, each d ( p, q ) is an integer, and for distinct elements p, q in M ( S ) wehave d ( p, p ) = 1 and d ( p, q ) = 0 . Proof.
The equation (0.3) gives us(1.6) X p ∈ S u ( p ) δ p = u = X ( s,t ) ∈ S × S d ( s, t ) δ st = X p ∈ S X ( s,t ) ∈ S × Sst = p d ( s, t ) δ p MAHYA GHANDEHARI, HAMED HATAMI AND NICO SPRONK
Since st = p necessitates ( s, t ) ≥ ( p, p ), we examine the coefficient of δ p tofind(1.7) u ( p ) = X ( s,t ) ≥ ( p,p ) st = p d ( s, t )from which we obtain (a). In particular, if p ∈ M ( S ) we obtain an emptysum in (a) and find d ( p, p ) = 1. The equation (0.4) implies that δ q · D = D · δ q and hence we obtain(1.8) X ( s,t ) ∈ S × S d ( s, t ) δ qs ⊗ δ t = X ( s,t ) ∈ S × S d ( s, t ) δ s ⊗ δ tq . If q p then there is no s in S for which qs = p . Hence examining thecoefficient of δ p ⊗ δ q and δ q ⊗ δ p , respectively, in (1.8), yields(1.9) 0 = X t ≥ q d ( p, t ) and X s ≥ q d ( s, p ) = 0 . Hence we have established (b). In particular, if q, p ∈ M ( S ) we have anempty sum in (b), so d ( p, q ) = 0.We can see for any pair ( p, q ) with p = q , so p q or q p , that d ( p, q )is determined by coefficients ( s, t ) > ( p, q ). Hence by induction, using thecoeficients d ( p, p ) and d ( p, q ) for distict maximal p, q as a base, we obtain(c). For example, if q ∈ M ( S \ M ( S )), then (b) implies for every p > q that d ( p, q ) = − X t>q d ( p, t ) = − d ( p, p ) = − d ( q, p ) = − d ( p, q ) is an integer. (cid:3) Let us see how Lemma 1.2 allows us to compute the diagonal D of ℓ ( S )for a finite semilattice S . Step 1.
We inductively define(1.10) S = S, S = S \ M ( S ) , . . . , S k +1 = S k \ M ( S k )and we let n ( S ) = min { k : S k +1 = ∅ } , so S n ( S ) = { o } and S n ( S )+1 = ∅ . Step 2.
We label S = { s , s , . . . , s | S |− } in any manner for which i ≥ j and s i ∈ S k ⇒ s j ∈ S k . Thus, the elements of M ( S k ) comprise the last part of the list of S k for k = 1 , . . . , n ( S ). In particular, s = o and s | S |− ∈ M ( S ). Step 3.
The diagonal D will be represented by an | S |×| S | matrix [ D ] =[ d ( s i , s j )]. The lower rightmost corner will be the | M ( S ) |×| M ( S ) | identitymatrix. We can then proceed, using formulas (b) and (a) from the lemmaabove, to compute the remaining entries of the lower rightmost ( | M ( S ) | +1) × ( | M ( S ) | + 1) corner of [ D ], etc., until we are done. MENABILITY CONSTANTS 7
In order to describe certain semilattices S , we define the semilattice graph Γ( S ) = ( S, e ( S )), where the vertex set is S and the edge set is given byordered pairs e ( S ) = { ( s, t ) ∈ S × S : s > t and there is no r in S for which s > r > t } . To picture such a graph for a finite semilattice S it is helpful to describelevels. Let S , S , . . . , S n ( S ) be the sequence of ideals of S given in (1.10).For s in S we let the level of s be given by λ ( s ) = n ( S ) − k where s ∈ M ( S k ) . Note that for the power set semilattice P ( T ), λ ( σ ) = | σ | , the cardinalityof σ . However, this relation need not hold for a subsemilattice of P ( T ), asis evident from the Example 1.4, below. A 6-element, 4-level semilattice isillustrated in (2.7).We apply this algorithm to obtain the following examples. We denote,for a finite semilattice S , the amenability constantAM( S ) = AM( ℓ ( S )) = k D k = X ( s,t ) ∈ S × S | d ( s, t ) | where we recall the well-known isometric identification ℓ ( S ) ⊗ γ ℓ ( S ) ∼ = ℓ ( S × S ). Example 1.3.
Let L n = { , , , . . . , n } be a “linear” semilattice with oper-ation st = s ∧ t = min { s, t } . Then we obtain diagonal with ( n + 1) × ( n + 1) matrix [ D ] = − . . . − . . . ... . . . . . . . . . ... . . . −
10 0 . . . − . Hence
AM( L n ) = 4 n + 1 . Example 1.4.
Let F n = { o, s , . . . , s n } be the n + 1 element “flat” semilat-tice with multiplications s i s j = o if i = j . Then we obtain unit u = δ s + · · · + δ s n + (1 − n ) δ o and diagonal with ( n + 1) × ( n + 1) -matrix [ D ] = n + 1 − − . . . − − . . . − . . . ...... ... . . . . . . − . . . . Hence
AM( F n ) = 4 n + 1 . MAHYA GHANDEHARI, HAMED HATAMI AND NICO SPRONK
Example 1.5.
Let F n = { o, s , . . . , s n , } be the unitasation of F n , above.Then we obtain diagonal with ( n + 2) × ( n + 2) matrix [ D ] = n − n + 2 − n . . . − n n − − n . . . − ... ... . . . ... ... − n . . . − n − − . . . − . Hence
AM( F n ) = 4 n + 4 n + 1 . The next example is less direct than the previous ones, so we offer a proof.
Example 1.6.
Let P n = P ( { , . . . , n } ) with multiplication st = s ∩ t . Thenthe diagonal D has n × n matrix which is, up to permutative similarity, theKronecker product (cid:20) − − (cid:21) ⊗ · · · ⊗ (cid:20) − − (cid:21) ( n times ) . Hence
AM( P n ) = 5 n . Proof. If s ∈ P n let χ s : { , . . . , n } → { , } = L be its indicator func-tion. It is easily verified that the map s χ s : P n → L n is a semi-lattice isomorphism. Thus there is an isometric identification ℓ ( P n ) ∼ = ℓ ( L ) ⊗ γ · · · ⊗ γ ℓ ( L ). Then it follows from Proposition 0.2 above that D = D ⊗ · · · ⊗ D where D is the diagonal for ℓ ( L ), which, by thealgorithm has matrix [ D ] = (cid:20) − − (cid:21) . The amenability constant AM( P n ) can be easily computed by induction. (cid:3) We have the following summary result.
Theorem 1.7. If S is a finite semilattice, then AM( S ) = 4 n + 1 for someinteger n ≥ . All such numbers are achieved. Proof.
We first establish that for p in S , d ( p, p ) ≥
0. This does not seemobvious from Lemma 1.2. We use a calculation from [1, §
3] which exploitsthe M¨obius function. We have that Σ : ℓ ( S ) → ℓ ∞ ( S ) is invertible and˜ D = P r ∈ S χ r ⊗ χ r is the diagonal for ℓ ∞ ( S ). Thus, using (1.3), we havethat D = Σ − ⊗ Σ − ( ˜ D ) = X r ∈ S X s ∈ S ˜ µ ( s, r ) δ s ! ⊗ X t ∈ S ˜ µ ( t, r ) δ t ! = X ( s,t ) ∈ S × S X r ∈ S ˜ µ ( s, r )˜ µ ( t, r ) ! δ s ⊗ δ t MENABILITY CONSTANTS 9 is the diagonal for ℓ ( S ), where ˜ µ ( s, t ) = µ ( s, t ) if s ≤ t and ˜ µ ( s, t ) = 0,otherwise. Inspecting the coeficient of δ p ⊗ δ p we obtain(1.11) d ( p, p ) = X r ∈ S ˜ µ ( p, r ) ≥ > µ ( p, p ) = µ ( p, p ) = 1 by [15, § X ( s,t ) ∈ S × S d ( s, t ) = X p ∈ S X ( s,t ) ∈ S × Sst = p d ( s, t ) = X p ∈ S u ( p ) = 1 . By symmetry, if p = q then | d ( p, q ) | + | d ( q, p ) | ≡ d ( p, q ) + d ( q, p ) mod 4.Hence we haveAM( S ) ≡ X ( s,t ) ∈ S × S | d ( s, t ) | ≡ X ( s,t ) ∈ S × S d ( s, t ) ≡ . Finally, Examples 1.3 and 1.4 provide us with semilattices admitting amenabil-ity constants 4 n + 1, for each integer n ≥ (cid:3) We now gain a crude lower bound for AM( S ) which we will require forProposition 2.4. Corollary 1.8.
For any finite semilattice S we have AM( S ) ≥ | S | − . Proof.
We have from (1.11) that d ( p, p ) ≥ p in S . It then followsfrom (1.9) that for p > o we have P t ≥ o d ( p, t ) = 0 from which we obtain P t = p | d ( p, t ) | ≥
1. It then follows thatAM( S ) = X ( s,t ) ∈ S × S | d ( s, t ) | ≥ d ( o, o ) + X p> d ( p, p ) + X t = p | d ( p, t ) | ≥ | S | − (cid:3) We note that if S is unital, then for p < u ( p ) = 0 and since d ( s, t ) = d ( t, s ) for ( s, t ) > ( p, p ) we find from Lemma 1.2 (a) that d ( p, p ) is even; inparticular d ( p, p ) ≥
2. The proof above may be adapted to show AM( S ) ≥ | S | −
3, in this case. We conjecture the estimate AM( S ) ≥ | S | − S .2. Banach algebras graded over semilattices
A Banach algebra A is graded over a semigroup S if we have closed sub-spaces A s for each s in S such that A = ℓ - M s ∈ S A s and A s A t ⊂ A st for s, t in S. We will be interested strictly in the case where S is a finite semilattice.Notice in this case each A s is a closed subalgebra of A . The next propo-sition can be proved by a straightforward adaptation of the proof of [14,Proposition 3.1]. However, we offer another proof. Proposition 2.1.
Let S be a finite semilattice and A be graded over S .Then A is amenable if and only if each A s is amenable. Proof.
Suppose A is amenable. If s ∈ S , then A s = L t ≤ s A t is an idealin A which is complemented and hence an amenable Banach algebra (see[12, Theorem 2.3.7], for example). It is easy the check that the projection π s : A s → A s is a quotient homomorphism. Hence it follows that if ( D sα )is an approximate diagonal for A s then (cid:0) π s ⊗ π s ( D sα ) (cid:1) is an approximatediagonal for A s . (This is quotient argument is noted in [12, Corollary 2.3.2]and [3, Proposition 2.5].)Now suppose that each A s is amenable. Let S , S , . . . , S n ( S ) be the se-quence of ideals from (1.10). For each n = 0 , , . . . , n ( S ) we set A n = L s ∈ S n A s and observe, for each n = 0 , , . . . , n ( S ) −
1, that we have anisometrically isomorphic identification A n / A n +1 = ℓ - M s ∈ M ( S n ) A s where multiplication in the latter is pointwise, i.e. A s A t = { } if s = t in M ( S n ). The pointwise algebra ℓ - L s ∈ M ( S n ) A s is amenable as each A s is amenable; if ( D s,α ) is a bounded approximate diagonal for each A s ,then in ℓ - M s ∈ M ( S n ) A s ⊗ γ ℓ - M s ∈ M ( S n ) A s ∼ = ℓ - M ( s,t ) ∈ M ( S n ) × M ( S n ) A s ⊗ γ A t the net of elements D α = P s ∈ M ( S n ) D s,α is an approximate diagonal. Thusif A n +1 is amenable, then A n must be too by [12, Theorem 2.3.10]. Thealgebra A n ( S ) = A o is amenable, and hence we may finish by an obviousinduction. (cid:3) In the computations which follow, we will require one of the following linking assumptions which are very natural for our examples. (LA1)
For each s in S there is a bounded approximate identity( u s,α ) α in A s , such that for each t ≤ s and a t ∈ A t we havelim α u s,α a t = a t = lim α a t u s,α . (LA2) For each s ∈ S there is a contractive character χ s : A s → C such that for each s, t in S , a s ∈ A s and a t ∈ A t , we have χ st ( a s b t ) = χ s ( a s ) χ t ( a t ).Notice that in (LA1), each ( u s,α ) α is a bounded approximate identity for A s = ℓ - L t ≤ s A t . Thus since A s is an A s -module, Cohen’s factorisation MENABILITY CONSTANTS 11 theorem [6, 32.22] tells us that(2.1) for each a in A s there is v s ∈ A s and a ′ in A s such that a = v s a ′ . There is a right factorisation analogue, and the result also holds on each A s module A t , where t ≤ s . We note that (LA2) is equivalent to having acontractive character χ : A → C such that χ | A s = χ s for each s .We note that many natural Banach algebras, graded over semilattices,which arise in harmonic analysis, satisfy (LA2). However, (LA1) can be usedwhenever each component algebra A s admits no characters. For example, ifwe have a (finite unital) semilattice S , a family of algebras {A s } s ∈ S each hav-ing no characters, and a system { η st : s, t ∈ S, s ≥ t } of homomorphisms, wecan make ℓ - L s ∈ S A s into a Banach algebra by setting a s a t = η sst ( a s ) η tst ( a t )for a s in A s and a t in A t . (This construction is analagous to that of the Clif-ford semigroup algebras which will be presented in Section 2.1, below.)This brings us to the main result of this article. Theorem 2.2.
Let A be a Banach algebra graded over a finite semilattice S such that each A s is amenable. If we have either that (LA1) holds, or(LA2) holds, then AM( A ) ≥ AM( S ) . Proof. A is amenable by the proposition above.Let us suppose (LA1) holds. We let for each p in S , π p : A → A p thecontractive projection. We define for a, b ∈ A , π p ( a ⊗ b ) = π p ( a ) ⊗ b and( a ⊗ b ) π p = a ⊗ π p ( b ). Clearly these actions extend linearly and continuouslyto define π p D and Dπ p for any D ∈ A ⊗ γ A .We let ( D α ) be a bounded approximate diagonal for A and D = X ( s,t ) ∈ S × S d ( s, t ) δ s ⊗ δ t be the unique diagonal for ℓ ( S ). We will prove that for p, q ∈ S and a ∈ A p , b ∈ A q that( ⋆ ) lim α am ( π p D α π q ) b = d ( p, q ) ab. This requires induction and we will need some preliminary steps.Suppose that q = p in S , say q p . If v q ∈ A q then (0.2) implies that(2.2) lim α π p ( D α · v q ) π q = lim α π p ( v q · D α ) π q = 0 . We note that on an elementary tensor in
A ⊗ A we have(2.3) m ( π p ( a ⊗ b · v q ) π q ) = X t ≥ q π p ( a ) π t ( b ) v q = X t ≥ q m ( π p ( a ⊗ b ) π t ) v q Now if b ∈ A q we find v q ∈ A q and b ′ in A q such that b = v q b ′ by (2.1). Wethen have, in analogy to Lemma 1.2 (b), using (2.2) and (2.3)(b ’) lim α X t ≥ q m ( π p D α π t ) b = lim α m (cid:0) π p ( D α · v q ) π q (cid:1) b ′ = 0 · b ′ = 0 . Similarly we see(b ’) lim α X s ≥ q bm ( π s D α π p ) = 0 . Note that if p, q ∈ M ( S ) with p = q , then then (b ’) takes the formlim α m ( π p D α π q ) b = 0 = d ( p, q ) b and a simlar version holds for (b ’). Thus ( ⋆ ) holds in this case.Now we show that for p ∈ S and b in A p that(2.4) lim α π p (cid:0) m ( D α ) (cid:1) b = u ( p ) b where u = P p ∈ S u ( p ) δ p is the unit for ℓ ( S ). By (2.1) there are v p in A p and b ′ in A p such that b = v p b ′ . We have that v p = lim α m ( D α ) v p = lim α X s ∈ S π s (cid:0) m ( D α ) (cid:1) v p = lim α X s ≥ p π s (cid:0) m ( D α ) (cid:1) v p + X s p π s (cid:0) m ( D α ) (cid:1) v p from which it follows thatlim α X s ≥ p π s (cid:0) m ( D α ) (cid:1) v p = lim α π p (cid:0) m ( D α ) v p (cid:1) = v p and hence(2.5) lim α X s ≥ p π s (cid:0) m ( D α ) (cid:1) b = lim α X s ≥ p π s (cid:0) m ( D α ) (cid:1) v p b ′ = v p b ′ = b. In particular, if p ∈ M ( S ), thenlim α π p (cid:0) m ( D α ) (cid:1) b = b = u ( p ) b. Then the equation (2.4) follows inductively from (2.5) and (1.4), using thecase of maximal p as a base.Now we establish an analogue of Lemma 1.2 (a). For an elementary tensor a ⊗ b in A ⊗ A , we have(2.6) π p ( ab ) = X ( s,t ) ≥ ( p,p ) st = p π s ( a ) π t ( b ) = X ( s,t ) ≥ ( p,p ) st = p m (cid:0) π s ( a ⊗ b ) π t (cid:1) . It then follows from (2.4) and (2.6) that for b ∈ A p (a’) u ( p ) b = lim α X ( s,t ) ≥ ( p,p ) st = p m ( π s D α π t ) b. Note that if p ∈ M ( S ), then by Proposition 1.1, (a’) becomes d ( p, p ) b = b = lim α m ( π p D α π p ) b. Thus ( ⋆ ) holds in this case. MENABILITY CONSTANTS 13
We now prove ( ⋆ ) by induction on pairs ( p, q ) in S × S with pairs ( p, q ) ∈ M ( S ) × M ( S ) as a base. If p ∈ S , the induction hypothesis is that for a, b ∈ A p lim α am ( π s D α π t ) b = d ( s, t ) ab for ( s, t ) > ( p, p ) with st = p. Notice that in the hypothesis above we have A p ⊂ A s ∩ A t , and, moreover,either t s or s t . But then it follows from (a’) and Lemma 1.2 (a) thatlim α am ( π p D α π p ) b = u ( p ) − X ( s,t ) > ( p,p ) st = p d ( s, t ) ab = d ( p, p ) ab which establishes ( ⋆ ) in this case. Also, if q = p , say q p , then for a in A p and b in A q the induction hypothesis is thatlim α am ( π p D α π t ) b = d ( p, t ) ab for t > q. Combining this with (b ’) and Lemma 1.2 (b) we obtain the equation ( ⋆ )for this case. We can use (b ’) in place of (b ’) above, to acheive ( ⋆ ) with p and q interchanged.We now use ( ⋆ ) to finish the proof. Let for p, q in Sη ( p, q ) = sup a ∈A p ,b ∈A q k ab kk a k k b k . We note that our assumption (LA1) provides that η ( p, q ) >
0. For ε > a ε in A p and b ε in A q be so k a ε b ε kk a ε kk b ε k ≥ (1 − ε ) η ( p, q ). Then by ( ⋆ ) we have | d ( p, q ) | k a ε b ε k = lim α k a ε m ( π p D α π q ) b ε k ≤ lim inf α k a ε m ( π p D α π q ) k k b ε k≤ lim inf α k a ε m ( π p D α π q ) kk a ε k k m ( π p D α π q ) k k a ε k k b ε k k m ( π p D α π q ) k≤ η ( p, q ) k a ε k k b ε k lim inf α k m ( π p D α π q ) k which implies(1 − ε ) | d ( p, q ) | ≤ lim inf α k m ( π p D α π q ) k ≤ lim inf α k π p D α π q k γ . ThusAM( S ) = X ( p,q ) ∈ S × S | d ( p, q ) | ≤ X ( p,q ) ∈ S × S lim inf α k π p D α π q k γ ≤ lim inf α X ( p,q ) ∈ S × S k π p D α π q k γ ( † ) = lim inf α k D α k ≤ sup α k D α k γ where the equality ( † ) holds because of the isometric identification A ⊗ γ A = ℓ - M s ∈ S A s ! ⊗ γ ℓ - M t ∈ S A t ! ∼ = ℓ - M ( s,t ) ∈ S × S A s ⊗ γ A t . Thus we have finished the case where we assumed (LA1).Now suppose we have (LA2). The mapΠ :
A → ℓ ( S ) , Π( a ) = X s ∈ S χ s (cid:0) π s ( a ) (cid:1) δ s is a contractive homomorphism. Hence it follows that if ( D α ) is a boundedapproximate diagonal for A then (cid:0) Π( D α ) (cid:1) is an approximate diagonal for ℓ ( S ). Thus the limit point, i.e. unique cluster point, D of (cid:0) Π( D α ) (cid:1) satisfies k D k γ = AM( S ), whence sup α k D α k γ ≥ lim α k Π( D α ) k γ ≥ AM( S ). (cid:3) It might seem plausible that in the situation of the theorem above, if itwere the case that AM( A s ) = 1, for each s , then AM( A ) = AM( S ). Indeedthis phenomenon was observed for S = L , in a special case in [14, Theorem2.3]. However this does not seem to hold in general, as we shall see below.2.1. Clifford semigroup algebras.
Let S be a semilattice, and for each s in S suppose we have a group G s , and for each t ≤ s a homomorphism η st : G s → G t such that for r ≥ s ≥ t in S we have η ss = id G s and η rs ◦ η st = η rt then G = F s ∈ S G s (disjoint union) admits a semigroup operation given by x s y t = η sst ( x s ) η tst ( y t )for x s in G s and y t in G t . It is straightforward to check that G is a semi-group, and is called a Clifford semigroup , as such a semigroup was first de-scribed in [2]. We note that the set of idempotents E ( G ) is { e s } s ∈ S , where e s is the neutral element of G s , and E ( G ) is a subsemigroup, isomorphic to S . It is clear that ℓ ( G ) = ℓ - M s ∈ S ℓ ( G s )and that ℓ ( G ) is thus graded over S . Note that ℓ ( G ) satisfies (LA1) bydesign, and satisfies (LA2) where the augmentation character is used oneach ℓ ( G s ). As with semilattices we will write AM( G ) = AM( ℓ ( G ))Consider the semilattice S = { o, s , s , s , s , } whose graph is givenbelow.(2.7) 1 |||||||| BBBBBBBB s |||||||| BBBBBBBB s (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) s QQQQQQQQQQQQQQQ s o MENABILITY CONSTANTS 15
Using the algorithm following Lemma 1.2, with the semilattice ordered aspresented, we obtain diagonal D with matrix(2.8) [ D ] = − − − − − − − − − − − −
11 0 0 − − . Thus we obtain amenability constant AM( S ) = 41.Now let n ≥ G n be the Clifford semigroup gradedover S for which G n,s = { e , a, . . . , a n − } and G n,s i = { e i } for all i = 3and all connecting homomorphisms are trivial. Here, { e , a, . . . , a n − } isa cyclic group, and each other { e i } is the trivial group. This is a finitedimensional commutative amenable algebra, and hence admits a unique di-agonal by Proposition 0.1. It is straightforward to verify that if we order thesemigroup { o, e , e , e , a, . . . , a n − , e , } we obtain matrix for the diagonal − − − n ) /n /n . . . /n − − − /n − /n . . . − /n − − /n − /n . . . − /n − n ) /n − /n − /n ( n + 1) /n . . . − /n − /n − /n . . . /n ... ... ... ... . . . . . . ... ... /n − /n − /n /n − . . . −
11 0 0 − . . . − . Notice that values in positions ( o, e ) , . . . , ( o, a n − ) sum to 0, the value inthe ( o, s ) position in (2.8) above. Similar results holds for all submatiriceswith indicies from { e , a, . . . , a n − } . Summing absolute values of all entriesin the matrix we obtain amenability constant AM( G n ) = 41 + 4( n − /n .Thus AM( G n ) = 41 + 4 n − n >
41 = AM( S ) . The constant AM( G ) = 43 is the smallest amenability constant we can findfor an commutative semigroup which is not of the form 4 n + 1.2.2. Algebras graded over linear semilattices.
We note that if G is afinite Clifford semigroup, graded over a linear semilattice L n , then AM( G ) =AM( L n ) = 4 n + 1. Indeed, this holds more generally, by the followingproposition. Proposition 2.3. If A = ℓ - L k ∈ L n A k is a graded Banach algebra whichsatisfies (LA1), and A k is contractible with AM( A k ) = 1 for each k in L n ,then AM( A ) = 4 n + 1 . Proof.
We have from Theorem 2.2 that AM( A ) ≥ AM( L n ) = 4 n + 1, henceit suffices to exhibit a diagonal D with k D k γ ≤ n + 1. We will show thatsuch D exists by induction.Write L n = { , , . . . , n } . We identify L k as an ideal of L n for each k = 0 , , . . . , n − u k,α ) is a boundedapproximate identity for A k , which satisfies (LA1), then the unit e k of A k is the limit point of ( u k,α ), and hence e k is the unit for A k = ℓ - L j ∈ L k A j .Note, moreover, that the assumption that AM( A k ) = 1 forces k e k k = 1.Let ε >
0. Suppose for k < n we have a diagonal D k for A k with (cid:13)(cid:13) D k (cid:13)(cid:13) γ < k + 1 + ε . For k = 0, such a diagonal exists as AM( A ) = 1. Welet D k +1 = ∞ X i =1 a i ⊗ b i , a i , b i ∈ A k +1 be a diagonal for A k +1 with k D k +1 k γ ≤ P ∞ i =1 k a i k k b i k < ε . We then set D k +1 = ∞ X i =1 a i · (cid:0) ( e k +1 − e k ) ⊗ ( e k +1 − e k ) + D k (cid:1) · b i . Clearly (cid:13)(cid:13)(cid:13) D k +1 (cid:13)(cid:13)(cid:13) γ ≤ (4 + (4 k + 1 + ε ))(1 + ε ) = 4( k + 1) + 1 + O ( ε ) . Applying the multiplication map, and noting that m ( D k ) = e k , we have m ( D k +1 ) = ∞ X i =1 a i (cid:0) e k +1 − e k − e k + e k + m ( D k ) (cid:1) b i = ∞ X i =1 a i e k +1 b i = m ( D k +1 ) = e k +1 so (0.3) for D k +1 is satisfied. Now if a ∈ A k +1 then by property (0.4) for D k +1 we have P ∞ i =1 ( aa i ) ⊗ b i = P ∞ i =1 a i ⊗ ( b i a ), so it follows that a · D k +1 = D k +1 · a . Now if a ∈ A k , then each aa i ∈ A k so a · D k +1 = ∞ X i =1 ( aa i ) · (cid:0) ( e k +1 − e k ) ⊗ ( e k +1 − e k ) + D k (cid:1) · b i = ∞ X i =1 (cid:0) [ aa i ( e k +1 − e k )] ⊗ ( e k +1 − e k ) + ( aa i ) · D k (cid:1) · b i = ∞ X i =1 D k · ( aa i b i ) = D k · a = a · D k MENABILITY CONSTANTS 17 which, by symmetric argument, is exactly the value of D k +1 · a . Since any a ∈ A k +1 is a sum a = π k +1 ( a ) + ( a − π k +1 ( a )) where, π k +1 ( a ) ∈ A k +1 and a − π k +1 ( a ) ∈ A k , we obtain (0.4) for D k +1 . (cid:3) We note that to generalise our proof of the preceding result to amenablebut not contractible Banach algebras, we would require at each stage ap-proximate diagonals D kα such that (cid:13)(cid:13) m ( D kα ) (cid:13)(cid:13) = 1, which we do not knowhow to construct, in general. We point the reader to [13, Theorem 2.3] tosee a computation performed on a Banach algebra graded over L .We note that we can modify the proof of Proposition 2.3 to see that aBanach algebra A = ℓ - L s ∈ F A s graded over F , where each A s is con-tractible with AM( A s ) = 1 , satisfies AM( A ) ≤
45. This is larger thanAM( F ) = 25 from Example 1.4. We have found no examples of suchBanach algebras A with AM( A ) >
25. However, we conjecture only forsemilattices S = L n , that a Banach algebra A = ℓ - L s ∈ S A s graded over S , where each A s is amenable with AM( A s ) = 1 , satisfies AM( A ) = AM( S ) . It would be interesting to find non-linear unital semilattices over which thisconjecture holds.2.3.
On allowable amenability constants.
We close by partially answer-ing a question posed in [3]. There it is proved, that there is no semigroup G such that 1 < AM( G ) <
5. It is further conjectured that there are nosemigroups G for which AM( G ) ∈ (5 , ∪ (7 , G with AM( G ) = 7. For commutativesemigroups there is a further gap. Proposition 2.4.
There is no commutative semigroup G such that < AM( G ) < . Proof.
Since G is commutative, it is proved in [5, Theorem 2.7] that if ℓ ( G )is amenable, then G is a Clifford semigroup, whose component groups areabelian, graded over a finite semilattice S . If AM( G ) <
9, then by Theorem2.2 then AM( S ) < | S | − ≤ AM( S ) ≤ | S | ≤
3. Clearly, if | S | = 1, S = L , and if | S | = 2, S = L . If | S | = 3 then S is either unital, in which case S = L , or S has 2 maximalelements, in which case S = F ; in either case AM( S ) = 9, contradictingour assumptions. Thus S = L or L . But it then follows by a straighfor-ward adaptation of [13, Theorem 2.3] that AM( G ) = 1 or 5. In particularAM( G ) ≤ (cid:3) Acknowledgements.
The authors are grateful to H.G. Dales for valu-able questions and discussion, and the Y. Choi for providing a preprint ofhis article [1].
References [1] Y. Choi. Biflatness of ℓ -semilattice algebras. To appear in Semigroup Forum , seeArXiv math.FA/0606366 .[2] A. H. Clifford. Semigroups admitting relative inverses.
Ann. Math. , 42:1037–1049,1941.[3] H. G. Dales, A. T.-M. Lau, and D. Strauss. Banach algebras on semigroups and theircompactifications. Manuscript, 2006.[4] J. Duncan and I. Namioka. Amenability of inverse semigroups and their semigroupalgebras.
Proc. Roy. Soc. Edinburgh Sect. A , 80:309–321, 1978.[5] N. Grønbæk. Amenability of discrete convolution algebras, the commutative case.
Pacific J. Math. , 143:243–249, 1990.[6] E. Hewitt and K. A. Ross.
Abstract Harmonic Analysis II , volume 152 of
Grundlehernder mathemarischen Wissenschaften . Springer, New York, 1970.[7] M. Ilie and N. Spronk. The spine of a Fourier-Stieltjes algebra.
Proc. London Math.Soc. (3) , 94:273–301, 2004.[8] M. Ilie and N. Spronk. The algebra generated by idempotents in a Fourier-Stieltjesalgebra. To appear in
Houston Math. J. , see Arxiv math.FA/0510514 , 2005.[9] J. Inoue. Some closed subalgebras of measure algebras and a generalization of P.J.Cohen’s theorem.
J. Math. Soc. Japan , 23:278–294, 1971.[10] B. E. Johnson. Approximate diagonals and cohomology of certain annihilator Banachalgebras.
AMer. J. Math , 94:685–698, 1972.[11] B. E. Johnson. Non-amenability of the Fourier algebra of a compact group.
J. LondonMath. Soc. , 50:361–374, 1994.[12] V. Runde.
Lectures on Amenability , volume 1774 of
Lec. Notes in Math.
Springer,Berlin Heildelberg, 2002.[13] V. Runde and N. Spronk. Operator amenability of Fourier-Stieltjes algebras.
Math.Proc. Camb. Phil. Soc. , 136:675–686, 2004.[14] V. Runde and N. Spronk. Operator amenability of Fourier-Stieltjes algebras II. Toappear in
Bull. London Math. Soc. , see ArXiv math.FA/0507373 , 2005.[15] R.P. Stanley.
Enumerative Combinatorics , volume I. Wadsworth & Brooks, Belmont,California, 1986.[16] R. Stokke. Approximate diagonals and følner conditions for amenable group andsemigroup algebras.
Studia. Math. , 164:139–159, 2004.[17] J. L. Taylor.
Measure Algebras , volume 16 of
Conference Board of Mathematical Sci-ences . American Mathematical Society, Providence, RI, 1973.
Mahya GhandehariAddress:
Department of Pure Mathematics, University of Wa-terloo, Waterloo, ON N2L 3G1, Canada
E-mail: [email protected]
Hamed HatamiAddress:
Department of Computer Science, University of To-ronto, Toronto, ON M5S 3G4, Canada
E-mail: [email protected]
Nico SpronkAddress:
Department of Pure Mathematics, University of Wa-terloo, Waterloo, ON N2L 3G1, Canada