View source for Cohomology of Banach algebras

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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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.E. Johnson,   "Cohomology of Banach algebras"  ''Mem. Amer. Math. Soc.'' , '''127'''  (1972)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.Ya. [A.Ya. Khelemskii] Helemsky,   "Cohomology of Banach and topological spaces" , Reidel  (Forthcoming)  (Translated from Russian)</TD></TR></table>