# Cohomology of Banach algebras

The groups , , where is a two-sided Banach module over a Banach algebra , defined as the cohomology groups of the cochain complex

the -dimensional chains of which are the continuous -linear operators from into , and

The cohomology of Banach algebras can also be introduced via a Banach analogue of the functor , and there is also an axiomatic definition.

Analogously to the cohomology of algebras, the elements of the one-dimensional cohomology group of a Banach algebra can be interpreted as continuous derivations from into modulo inner derivations, and the elements of the two-dimensional cohomology group can be interpreted as equivalence classes of extensions of by in which is complemented. At the same time a number of specific analytic and topological concepts can be expressed in the language of cohomology of Banach algebras.

An algebra such that for all is said to be completely separable; these algebras are characterized by the fact that all their extensions split. The specific character of Banach structures is reflected by the fact that such a requirement is very rigid: A completely-separable commutative Banach algebra necessarily has finite spectrum (space of maximal ideals). In particular, a completely-separable function algebra is the direct sum of finitely many copies of (the field of complex numbers).

The class of Banach algebras with trivial cohomology in higher dimension is not so restricted; it contains, e.g., the class of biprojective algebras, i.e. algebras that are projective as two-sided Banach -modules. The -algebra and the -algebra of a compact group are biprojective, as are the algebras of nuclear operators in all classical Banach spaces. Under certain conditions on the Banach structure, topologically-simple biprojective algebras can be characterized completely, and every semi-simple biprojective algebra is a topological direct sum of such algebras.

A commutative algebra is said to be weakly hereditary if its maximal ideals are projective. This property is equivalent to the triviality of for those for which for all , . For an ideal in a commutative Banach algebra to be projective it is necessary that its spectrum be paracompact. If this condition is also sufficient. In particular, is weakly hereditary if and only if all sets of the form , , are paracompact.

The space dual to a two-sided -module is itself a two-sided -module. Algebras with for all and are called amenable, since for this property is equivalent to the amenability (averageability) of . In general, is amenable if and only if the algebra

has a bounded approximate identity.

#### References

 [1] B.E. Johnson, "Cohomology of Banach algebras" Mem. Amer. Math. Soc. , 127 (1972) [2] A.Ya. Khelemskii, "Lower values that admit the global homological dimension of Banach function algebras" Trudy Sem. Petrovsk. : 3 (1978) pp. 223–242 (In Russian)