Projections and Approximate Identities for Ideals in Group Algebras

Amenable and Weakly Amenable Banach Algebras with Compact Multiplication

“…The only previously known examples of weakly amenable, radical, commutative Banach algebras were given in [11], and were obtained as quotients I(E)ÂJ(E) for certain sets E not of synthesis in L 1 (R). A direct attempt to obtain an amenable example in this manner will fail if I is a closed amenable ideal in L 1 (G) for some locally compact abelian group G, then hull(I ) is necessarily of synthesis, [23]. We show that under suitable conditions our construction yields an integral domain, in which case our example is of necessity not amenable, and is different from the the earlier example.…”

The continuity of derivations from group algebras: factorizable and connected groups

“…That is to ensure that finite codimensional ideals in £ l {H k ) have bounded approximate units so that Cohen's factorization theorem can be applied. In order to ensure this, it would suffice, in the statement of Lemma 2.1, to suppose only that the subgroups H k are amenable, see [12]. However the abelian case will suffice for the automatic continuity proofs and it is not obvious that the lemmas would apply in significantly greater generality if 'abelian' were replaced by 'amenable'.…”

“…Such factorization results are an example of the sort of information about the structure of group algebras which is required to answer the automatic continuity question. In the cases where G is abelian or compact, finite codimensional ideals in c\G) have bounded approximate units because in these cases G is amenable, see [12], and the required factorizations follow from Cohen's theorem, see [ 2,Theorem 11.10]. However, many factorizable and connected groups are not amenable and other methods have to be used to prove the required factorization results.…”

On primary ideals in group algebras

“…Clearly [6,Theorem]. Since H has polynomial growth and a symmetric Ll-algebra, the injection theorem [7,Theorem 3.2] implies that F is a spectral set for H. Moreover, since H is amenable and k(F) has finite codimension in LI(H), k(F) has bounded approximate units [12,Theorem 2]. Consider now E = {ker ~; z E (G ~) ^, ker T t H ~ ker z}.…”