Proposition 12 (Ceccherini-Silberstein, Machi and Scarabotti [7]) Let G be an amenable group and let A be a CA on G.

Lemma 19 (Step 1 in proof of [7, Theorem 3]) Let G be a finitely generated amenable group, q [greater than or equal to] 2, and n > r > 0.

Let [pi]: G [right arrow] O(H) be an ergodic orthogonal representation of a finitely generated amenable group G, and let b: G [right arrow] H be a 1-cocycle associated to [pi].

Let G be finitely generated amenable group admitting a controlled Folner sequence.

We can also give a negative answer to Question 1.2 for some amenable groups. One ingredient for this is the following monotonicity property.

for which there is a Cayley graph that contains a copy of the infinite binary tree as a subgraph, contains in particular all non-amenable groups ([2, Theorem 1.5]), as well as all elementary amenable groups with exponential growth (by [4] such groups contain a free subsemigroup).

Our result in particular shows that if B is a commutative C*-algebra or a matrix space, then every continuous surjective zero products preserving map T : A(G) [right arrow] B is a weighted homomorphism where G is an amenable group. It also provides us an equivalent condition for T to be a homomorphism.

Let Gbea locally compact amenable group. The multiplier algebra of A(G) is B(G).

Let G be an inner amenable group and let A be a Borel subset of G.

A good deal of attention was paid to the study of inner amenable groups. The study of inner invariant means was initiated by Effros [5] and pursued by Akemann [1], Yuan [25] for discrete groups, Lau and Paterson [13] and Yuan [26] for locally compact groups, and Ling [15] and Mohammadzadeh and Nasr-Isfahani [18] for semigroups.

As corollaries, the amalgamated free product of an infinite amenable group and a residually finite group over a finite subgroup (for instance [SL.sub.3][(Z).sub.*A] G with any infinite amenable group G and a common finite subgroup A) is in A (Corollary 8); and the amalgamated free product of two amenable groups over a finite subgroup is also contained in A (Corollary 9).

Let H be an amenable group and let [pi] : G [??] H be a group epimorphism and let A < G be a subgroup such that [pi][|.sub.A] is injective and [H : [pi](A)] [greater than or equal to] 2.

Punch is a win-win, both for

amenable groups of friends who can agree on their cocktail order and for restaurants and bars that can batch it ahead of time for easy service.

Several of his appendices also offer independent interest, covering

amenable groups, Banach algebra, bundles of C*-algebras, groups, representations of C*-algebras, direct integrals, Effros's ideal center decomposition, the Fell topology.

Willis, Weak convergence is not strong for

amenable groups, Canadian Mathematical Bulletin, 44 (2)(2001), 231-241.