Approximate Diagonals and Cohomology of Certain Annihilator Banach Algebras

In order to summarize the contents of this paper it is necessary to use some of the definitions in 5. These definitions are repeated below at the beginning of Section 1. Although this paper adds to the material in 5 we have attempted to reduce the number of references to o to a minimum. An approximate diagonal in a Banach algebra 31 is a bounded net ma in 31 a for all a in 31 in the norm topologies where 31? 31 is the usual projective tensor product 2 and -k is the continuous linear map 31? 31->31 defined by ir (a? b) =ab. The net {m 31) ** with aM = Ma and? r** (M) a = a for all a? 31. In the algebraic cohomology of algebras the existence of an element m of A A with ma = am and ir (m) the identity of A is equivalent to the statement that H1 (A, X) =0 for all 31 modules X almost never holds? in fact, in the commutative case for all A bimodules X. Among Banach algebras the statement jEPSX) =0 (see 3; Theorem 1 and 5; Proposition 8. 1) we can always find a Banach 31 module? with S¹ (3I, 36) =tO unless 31 zz? n. The existence of an approxi? mate diagonal is not unusual and is equivalent to the statement !M* (31, 36*) = 0 for all Banach 31 modules X; thatis 31 is an emenable Banach algebra. The element M plays the same role in the theory of amenable tlgebras that the invariant mean does for amenable groups. The definitions used from 5 and the theorem connecting amenability with approximate diagonals are given in? 1. In? 2 we show that if 31 has a virtual diagonal with Ma? 31? St for all a in 3T then 913 (31, 3E)? . 0 for all Banach 31 modules 3? . Algebras satisfying this condition are grdup algebras of compact topological groups and the algebra c0 of convergent sequences of complex numbers. Although we do not prove it the result also applies to any annihilator B* algebra with only finite dimensional minimal ideals. In? 3 we show that if 31 is a semi simple commutative amenable Banach algebra then there is a Banach 31 module X with 3/2 (3I, 3E) ⁰, unless 31 is finite