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The groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c0231201.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c0231202.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c0231203.png" /> is a two-sided Banach module over a Banach algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c0231204.png" />, defined as the cohomology groups of the cochain complex
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c0231205.png" /></td> </tr></table>
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{{TEX|auto}}
 +
{{TEX|done}}
  
the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c0231206.png" />-dimensional chains of which are the continuous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c0231207.png" />-linear operators from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c0231208.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c0231209.png" />, and
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c02312010.png" /></td> </tr></table>
+
$$
 +
0 \rightarrow  C  ^ {0} ( A, X)  \rightarrow \dots \rightarrow  C  ^ {n} ( A, X)
 +
\rightarrow ^ { {\delta  ^ {n}} }  C ^ {n + 1 } ( A, X)  \rightarrow \dots ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c02312011.png" /></td> </tr></table>
+
the  $  n $-
 +
dimensional chains of which are the continuous  $  n $-
 +
linear operators from  $  A $
 +
into  $  X $,
 +
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c02312012.png" /></td> </tr></table>
+
$$
 +
\delta  ^ {n} f ( a _ {1} \dots a _ {n + 1 }  )  = \
 +
a _ {1} f ( a _ {2} \dots a _ {n + 1 }  ) +
 +
$$
  
The cohomology of Banach algebras can also be introduced via a Banach analogue of the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c02312013.png" />, and there is also an axiomatic definition.
+
$$
 +
+
 +
\sum _ {k = 1 } ^ { n }  (- 1)  ^ {k} f ( a _ {1} \dots a _ {k} a _ {k + 1 }  \dots a _ {n + 1 }  ) +
 +
$$
  
Analogously to the cohomology of algebras, the elements of the one-dimensional cohomology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c02312014.png" /> of a Banach algebra can be interpreted as continuous derivations from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c02312015.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c02312016.png" /> modulo inner derivations, and the elements of the two-dimensional cohomology group can be interpreted as equivalence classes of extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c02312017.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c02312018.png" /> in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c02312019.png" /> 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.
+
$$
 +
+
 +
(- 1) ^ {n + 1 } f ( a _ {1} \dots a _ {n} ) a _ {n + 1 }  .
 +
$$
  
An algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c02312020.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c02312021.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c02312022.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c02312023.png" /> (the field of complex numbers).
+
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.
  
The class of Banach algebras with trivial cohomology in higher dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c02312024.png" /> is not so restricted; it contains, e.g., the class of biprojective algebras, i.e. algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c02312025.png" /> that are projective as two-sided Banach <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c02312026.png" />-modules. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c02312027.png" />-algebra and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c02312028.png" />-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.
+
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.
  
A commutative algebra is said to be weakly hereditary if its maximal ideals are projective. This property is equivalent to the triviality of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c02312029.png" /> for those <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c02312030.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c02312031.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c02312032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c02312033.png" />. For an ideal in a commutative Banach algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c02312034.png" /> to be projective it is necessary that its spectrum be paracompact. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c02312035.png" /> this condition is also sufficient. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c02312036.png" /> is weakly hereditary if and only if all sets of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c02312037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c02312038.png" />, are paracompact.
+
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 space dual to a two-sided <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c02312039.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c02312040.png" /> is itself a two-sided <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c02312041.png" />-module. Algebras with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c02312042.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c02312043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c02312044.png" /> are called amenable, since for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c02312045.png" /> this property is equivalent to the amenability (averageability) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c02312046.png" />. In general, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c02312047.png" /> is amenable if and only if the algebra
+
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.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023120/c02312048.png" /></td> </tr></table>
+
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.
 
has a bounded approximate identity.
Line 29: Line 106:
 
====References====
 
====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>
 
<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====
 
====Comments====
 
  
 
====References====
 
====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>
 
<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>

Revision as of 17:45, 4 June 2020


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)

Comments

References

[a1] A.Ya. [A.Ya. Khelemskii] Helemsky, "Cohomology of Banach and topological spaces" , Reidel (Forthcoming) (Translated from Russian)
How to Cite This Entry:
Cohomology of Banach algebras. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cohomology_of_Banach_algebras&oldid=46389
This article was adapted from an original article by A.Ya. Khelemskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article