Cohomology of Banach algebras

The groups $ H ^ {n} ( A, X) $, $ n \geq 0 $, where $ X $ is a two-sided Banach module over a Banach algebra $ A $, defined as the cohomology groups of the cochain complex

$$ 0 \rightarrow C ^ {0} ( A, X) \rightarrow \dots \rightarrow C ^ {n} ( A, X) \rightarrow ^ { {\delta ^ {n}} } C ^ {n + 1 } ( A, X) \rightarrow \dots , $$

the $ n $- dimensional chains of which are the continuous $ n $- linear operators from $ A $ into $ X $, and

$$ \delta ^ {n} f ( a _ {1} \dots a _ {n + 1 } ) = \ a _ {1} f ( a _ {2} \dots a _ {n + 1 } ) + $$

$$ + \sum _ {k = 1 } ^ { n } (- 1) ^ {k} f ( a _ {1} \dots a _ {k} a _ {k + 1 } \dots a _ {n + 1 } ) + $$

$$ + (- 1) ^ {n + 1 } f ( a _ {1} \dots a _ {n} ) a _ {n + 1 } . $$

The cohomology of Banach algebras can also be introduced via a Banach analogue of the functor $ \mathop{\rm Ext} $, and there is also an axiomatic definition.

Analogously to the cohomology of algebras, the elements of the one-dimensional cohomology group $ H ^ {1} ( X, A) $ of a Banach algebra can be interpreted as continuous derivations from $ A $ into $ X $ modulo inner derivations, and the elements of the two-dimensional cohomology group can be interpreted as equivalence classes of extensions of $ A $ by $ X $ in which $ X $ 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 $ A $ such that $ H ^ {2} ( A, X) = 0 $ for all $ X $ 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 $ \mathbf C $( the field of complex numbers).

The class of Banach algebras with trivial cohomology in higher dimension $ ( n \geq 3 ) $ is not so restricted; it contains, e.g., the class of biprojective algebras, i.e. algebras $ A $ that are projective as two-sided Banach $ A $- modules. The $ L _ {1} $- algebra and the $ C ^ {*} $- 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 $ H ^ {2} ( A, X) $ for those $ X $ for which $ xa = \lambda x $ for all $ x \in X $, $ a \in A $. For an ideal in a commutative Banach algebra $ A $ to be projective it is necessary that its spectrum be paracompact. If $ A = C ( \Omega ) $ this condition is also sufficient. In particular, $ C ( \Omega ) $ is weakly hereditary if and only if all sets of the form $ \Omega \setminus \{ t \} $, $ t \in \Omega $, are paracompact.

The space dual to a two-sided $ A $- module $ X $ is itself a two-sided $ A $- module. Algebras with $ H ^ {n} ( A, X ^ {*} ) = 0 $ for all $ X $ and $ n > 0 $ are called amenable, since for $ A = L _ {1} ( G) $ this property is equivalent to the amenability (averageability) of $ G $. In general, $ A $ is amenable if and only if the algebra

$$ I _ \Delta = \ \left \{ { u = \sum _ {k = 1 } ^ \infty a _ {k} \otimes b _ {k} \in A _ {t} \widehat \otimes A _ {t} } : { \sum _ {k = 1 } ^ \infty a _ {k} b _ {k} = 0 } \right \} $$

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)
[a1]	A.Ya. [A.Ya. Khelemskii] Helemsky, "Cohomology of Banach and topological spaces" , Reidel (Forthcoming) (Translated from Russian)