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A [[Normed algebra|normed algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a1106701.png" /> is said to be Arens regular if the pair of intrinsically defined Arens products (introduced by R. Arens in [[#References|[a1]]] and [[#References|[a2]]]; cf. [[Arens multiplication|Arens multiplication]]) on the double dual space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a1106702.png" /> are identical. Since both Arens products extend the product on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a1106703.png" /> (relative to the natural embedding mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a1106704.png" />), a [[Banach algebra|Banach algebra]] on a reflexive [[Banach space|Banach space]] (cf. also [[Reflexive space|Reflexive space]]) is Arens regular. S. Sherman has shown [[#References|[a10]]] that the double dual of a [[C*-algebra|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a1106705.png" />-algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a1106706.png" /> has a natural interpretation as the [[Von Neumann algebra|von Neumann algebra]] generated by the universal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a1106707.png" />-representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a1106708.png" />. Hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a1106709.png" />-algebras are always Arens regular.
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It is easy to show that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067010.png" /> is commutative under either Arens product, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067011.png" /> is Arens regular. The following fundamental result is due to J. Hennefeld [[#References|[a5]]], based on work of J.S. Pym [[#References|[a8]]] making use of Grothendieck's criterion for weak compactness.
+
{{TEX|auto}}
 +
{{TEX|done}}
  
The following conditions are equivalent for a Banach algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067012.png" />:
+
A [[Normed algebra|normed algebra]]  $  A $
 +
is said to be Arens regular if the pair of intrinsically defined Arens products (introduced by R. Arens in [[#References|[a1]]] and [[#References|[a2]]]; cf. [[Arens multiplication|Arens multiplication]]) on the double dual space  $  A ^ {* * } $
 +
are identical. Since both Arens products extend the product on  $  A $(
 +
relative to the natural embedding mapping  $  \kappa : A \rightarrow {A ^ {* * } } $),
 +
a [[Banach algebra|Banach algebra]] on a reflexive [[Banach space|Banach space]] (cf. also [[Reflexive space|Reflexive space]]) is Arens regular. S. Sherman has shown [[#References|[a10]]] that the double dual of a [[C*-algebra| $  C  ^ {*} $-
 +
algebra]]  $  A $
 +
has a natural interpretation as the [[Von Neumann algebra|von Neumann algebra]] generated by the universal  $  * $-
 +
representation of  $  A $.
 +
Hence  $  C  ^ {*} $-
 +
algebras are always Arens regular.
  
a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067013.png" /> is Arens regular;
+
It is easy to show that if  $  A ^ {* * } $
 +
is commutative under either Arens product, then  $  A $
 +
is Arens regular. The following fundamental result is due to J. Hennefeld [[#References|[a5]]], based on work of J.S. Pym [[#References|[a8]]] making use of Grothendieck's criterion for weak compactness.
  
b) for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067014.png" /> the adjoint <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067015.png" /> of the left regular representation is weakly compact;
+
The following conditions are equivalent for a Banach algebra  $  A $:
  
c) for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067016.png" /> the adjoint <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067017.png" /> of the right regular representation is weakly compact;
+
a) $  A $
 +
is Arens regular;
  
d) for any bounded sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067019.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067020.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067021.png" />, the iterated limits
+
b) for each  $  a \in A $
 +
the adjoint  $  {L _ {a}  ^ {*} } : {A  ^ {*} } \rightarrow {A  ^ {*} } $
 +
of the left regular representation is weakly compact;
  
<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/a/a110/a110670/a11067022.png" /></td> </tr></table>
+
c) for each  $  a \in A $
 +
the adjoint  $  {R _ {a}  ^ {*} } : {A  ^ {*} } \rightarrow {A  ^ {*} } $
 +
of the right regular representation is weakly compact;
 +
 
 +
d) for any bounded sequences  $  \{ a _ {n} \} _ {n \in \mathbf N }  $
 +
and  $  \{ b _ {n} \} _ {n \in \mathbf N }  $
 +
in  $  A $
 +
and any  $  \omega \in A  ^ {*} $,
 +
the iterated limits
 +
 
 +
$$
 +
{\lim\limits } _ { n } {\lim\limits } _ { m } \omega ( a _ {n} b _ {n} ) , \quad {\lim\limits } _ { m } {\lim\limits } _ { n } \omega ( a _ {n} b _ {n} )
 +
$$
  
 
are equal when they both exist.
 
are equal when they both exist.
Line 19: Line 53:
 
This theorem easily implies that subalgebras and quotient algebras (with respect to closed ideals) of Arens-regular algebras are Arens regular, as first noted in [[#References|[a3]]].
 
This theorem easily implies that subalgebras and quotient algebras (with respect to closed ideals) of Arens-regular algebras are Arens regular, as first noted in [[#References|[a3]]].
  
Arens regularity is rare among general Banach algebras. N.J. Young [[#References|[a11]]] has shown that for a locally compact group (cf. also [[Compact group|Compact group]]; [[Locally compact skew-field|Locally compact skew-field]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067024.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067025.png" />) is Arens regular if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067026.png" /> is finite. P. Civin and B. Yood had proved this for Abelian groups in [[#References|[a3]]]. In [[#References|[a12]]] it is shown that the measure algebra (cf. [[Algebra of measures|Algebra of measures]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067027.png" /> of a locally compact semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067028.png" /> in which multiplication is at least singly continuous is Arens regular if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067029.png" /> is. These are, in turn, equivalent to either:
+
Arens regularity is rare among general Banach algebras. N.J. Young [[#References|[a11]]] has shown that for a locally compact group (cf. also [[Compact group|Compact group]]; [[Locally compact skew-field|Locally compact skew-field]]) $  G $,  
 +
$  L  ^ {1} ( G ) $(
 +
or $  M ( G ) $)  
 +
is Arens regular if and only if $  G $
 +
is finite. P. Civin and B. Yood had proved this for Abelian groups in [[#References|[a3]]]. In [[#References|[a12]]] it is shown that the measure algebra (cf. [[Algebra of measures|Algebra of measures]]) $  M ( S ) $
 +
of a locally compact semi-group $  S $
 +
in which multiplication is at least singly continuous is Arens regular if and only if $  {\mathcal l}  ^ {1} ( S ) $
 +
is. These are, in turn, equivalent to either:
  
there do not exist sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067031.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067032.png" /> such that the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067034.png" /> are disjoint;
+
there do not exist sequences $  \{ u _ {n} \} $
 +
and $  \{ v _ {m} \} $
 +
in $  S $
 +
such that the sets $  \{ {u _ {n} v _ {m} } : {m > n } \} $
 +
and $  \{ {u _ {n} v _ {m} } : {m < n } \} $
 +
are disjoint;
  
the semi-group operation can be extended to the [[Stone–Čech compactification|Stone–Čech compactification]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067035.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067036.png" /> as a discrete space.
+
the semi-group operation can be extended to the [[Stone–Čech compactification|Stone–Čech compactification]] $  \beta S $
 +
of $  S $
 +
as a discrete space.
  
In [[#References|[a13]]], Young has proved that the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067037.png" /> of approximable operators (i.e., those uniformly approximable by finite-rank operators) on a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067038.png" /> is regular if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067039.png" /> is reflexive (cf. [[Reflexive space|Reflexive space]]). Hence, if the Banach algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067040.png" /> of all bounded linear operators on a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067041.png" /> is Arens regular, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067042.png" /> must be reflexive. He also shows that there are reflexive Banach spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067043.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067044.png" /> not Arens regular.
+
In [[#References|[a13]]], Young has proved that the algebra $  B _ {A} ( X ) $
 +
of approximable operators (i.e., those uniformly approximable by finite-rank operators) on a Banach space $  X $
 +
is regular if and only if $  X $
 +
is reflexive (cf. [[Reflexive space|Reflexive space]]). Hence, if the Banach algebra $  B ( X ) $
 +
of all bounded linear operators on a Banach space $  X $
 +
is Arens regular, then $  X $
 +
must be reflexive. He also shows that there are reflexive Banach spaces $  X $
 +
with $  B ( X ) $
 +
not Arens regular.
  
Á. Rodriguez-Palacios [[#References|[a9]]] has shown that any (even non-associative) continuous multiplication on a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067045.png" /> is Arens regular if and only if every bounded linear mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067046.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067047.png" /> is weakly compact (cf. [[Weak topology|Weak topology]]). <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067048.png" />-algebras satisfy this criterion.
+
Á. Rodriguez-Palacios [[#References|[a9]]] has shown that any (even non-associative) continuous multiplication on a Banach space $  A $
 +
is Arens regular if and only if every bounded linear mapping from $  A $
 +
into $  A  ^ {*} $
 +
is weakly compact (cf. [[Weak topology|Weak topology]]). $  C  ^ {*} $-
 +
algebras satisfy this criterion.
  
A weaker version of Arens regularity was introduced by M. Grosser [[#References|[a4]]]. An approximately unital [[Banach algebra|Banach algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067049.png" /> is said to be semi-regular if it satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067050.png" /> for all mixed identities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067051.png" />. (An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110670/a11067052.png" /> is called a mixed identity if it is simultaneously a right identity for the first Arens product and a left identity for the second Arens product, see [[Arens multiplication|Arens multiplication]].) He shows that an Arens-regular algebra is semi-regular and that any commutative approximately unital Banach algebra is semi-regular.
+
A weaker version of Arens regularity was introduced by M. Grosser [[#References|[a4]]]. An approximately unital [[Banach algebra|Banach algebra]] $  A $
 +
is said to be semi-regular if it satisfies $  R ^ {* * } ( e ) = L ^ {* * } ( e ) $
 +
for all mixed identities $  e $.  
 +
(An element $  e \in A ^ {* * } $
 +
is called a mixed identity if it is simultaneously a right identity for the first Arens product and a left identity for the second Arens product, see [[Arens multiplication|Arens multiplication]].) He shows that an Arens-regular algebra is semi-regular and that any commutative approximately unital Banach algebra is semi-regular.
  
 
The most comprehensive recent (1996) exposition is [[#References|[a7]]], which contains numerous further references.
 
The most comprehensive recent (1996) exposition is [[#References|[a7]]], which contains numerous further references.

Revision as of 18:48, 5 April 2020


A normed algebra $ A $ is said to be Arens regular if the pair of intrinsically defined Arens products (introduced by R. Arens in [a1] and [a2]; cf. Arens multiplication) on the double dual space $ A ^ {* * } $ are identical. Since both Arens products extend the product on $ A $( relative to the natural embedding mapping $ \kappa : A \rightarrow {A ^ {* * } } $), a Banach algebra on a reflexive Banach space (cf. also Reflexive space) is Arens regular. S. Sherman has shown [a10] that the double dual of a $ C ^ {*} $- algebra $ A $ has a natural interpretation as the von Neumann algebra generated by the universal $ * $- representation of $ A $. Hence $ C ^ {*} $- algebras are always Arens regular.

It is easy to show that if $ A ^ {* * } $ is commutative under either Arens product, then $ A $ is Arens regular. The following fundamental result is due to J. Hennefeld [a5], based on work of J.S. Pym [a8] making use of Grothendieck's criterion for weak compactness.

The following conditions are equivalent for a Banach algebra $ A $:

a) $ A $ is Arens regular;

b) for each $ a \in A $ the adjoint $ {L _ {a} ^ {*} } : {A ^ {*} } \rightarrow {A ^ {*} } $ of the left regular representation is weakly compact;

c) for each $ a \in A $ the adjoint $ {R _ {a} ^ {*} } : {A ^ {*} } \rightarrow {A ^ {*} } $ of the right regular representation is weakly compact;

d) for any bounded sequences $ \{ a _ {n} \} _ {n \in \mathbf N } $ and $ \{ b _ {n} \} _ {n \in \mathbf N } $ in $ A $ and any $ \omega \in A ^ {*} $, the iterated limits

$$ {\lim\limits } _ { n } {\lim\limits } _ { m } \omega ( a _ {n} b _ {n} ) , \quad {\lim\limits } _ { m } {\lim\limits } _ { n } \omega ( a _ {n} b _ {n} ) $$

are equal when they both exist.

This theorem easily implies that subalgebras and quotient algebras (with respect to closed ideals) of Arens-regular algebras are Arens regular, as first noted in [a3].

Arens regularity is rare among general Banach algebras. N.J. Young [a11] has shown that for a locally compact group (cf. also Compact group; Locally compact skew-field) $ G $, $ L ^ {1} ( G ) $( or $ M ( G ) $) is Arens regular if and only if $ G $ is finite. P. Civin and B. Yood had proved this for Abelian groups in [a3]. In [a12] it is shown that the measure algebra (cf. Algebra of measures) $ M ( S ) $ of a locally compact semi-group $ S $ in which multiplication is at least singly continuous is Arens regular if and only if $ {\mathcal l} ^ {1} ( S ) $ is. These are, in turn, equivalent to either:

there do not exist sequences $ \{ u _ {n} \} $ and $ \{ v _ {m} \} $ in $ S $ such that the sets $ \{ {u _ {n} v _ {m} } : {m > n } \} $ and $ \{ {u _ {n} v _ {m} } : {m < n } \} $ are disjoint;

the semi-group operation can be extended to the Stone–Čech compactification $ \beta S $ of $ S $ as a discrete space.

In [a13], Young has proved that the algebra $ B _ {A} ( X ) $ of approximable operators (i.e., those uniformly approximable by finite-rank operators) on a Banach space $ X $ is regular if and only if $ X $ is reflexive (cf. Reflexive space). Hence, if the Banach algebra $ B ( X ) $ of all bounded linear operators on a Banach space $ X $ is Arens regular, then $ X $ must be reflexive. He also shows that there are reflexive Banach spaces $ X $ with $ B ( X ) $ not Arens regular.

Á. Rodriguez-Palacios [a9] has shown that any (even non-associative) continuous multiplication on a Banach space $ A $ is Arens regular if and only if every bounded linear mapping from $ A $ into $ A ^ {*} $ is weakly compact (cf. Weak topology). $ C ^ {*} $- algebras satisfy this criterion.

A weaker version of Arens regularity was introduced by M. Grosser [a4]. An approximately unital Banach algebra $ A $ is said to be semi-regular if it satisfies $ R ^ {* * } ( e ) = L ^ {* * } ( e ) $ for all mixed identities $ e $. (An element $ e \in A ^ {* * } $ is called a mixed identity if it is simultaneously a right identity for the first Arens product and a left identity for the second Arens product, see Arens multiplication.) He shows that an Arens-regular algebra is semi-regular and that any commutative approximately unital Banach algebra is semi-regular.

The most comprehensive recent (1996) exposition is [a7], which contains numerous further references.

References

[a1] R. Arens, "Operations induced in function classes" Monatsh. Math. , 55 (1951) pp. 1–19
[a2] R. Arens, "The adjoint of a bilinear operation" Proc. Amer. Math. Soc. , 2 (1951) pp. 839–848
[a3] P. Civin, B. Yood, "The second conjugate space of a Banach algebra as an algebra" Pacific J. Math. , 11 (1961) pp. 847–870
[a4] M. Grosser, "Arens semiregular Banach algebras" Monatsh. Math. , 98 : 1 (1984) pp. 41–52
[a5] J.O. Hennefeld, "A note on the Arens products" Pacific J. Math. , 26 (1968) pp. 115–119
[a6] S. Kaijser, "On Banach modules I" Math. Proc. Cambridge Philos. Soc. , 90 : 3 (1981) pp. 423–444
[a7] T.W. Palmer, "Banach algebras and the general theory of -algebras I" , Encycl. Math. Appl. , 49 , Cambridge Univ. Press (1994)
[a8] J.S. Pym, "The convolution of functionals on spaces of bounded functions" Proc. London Math. Soc. (3) , 15 (1965) pp. 84–104
[a9] Á. Rodriguez-Palacios, "A note on Arens regularity" Quart. J. Math. Oxford Ser. (2) , 38 : 149 (1987) pp. 1991–1993
[a10] S. Sherman, "The second adjoint of a -algebra" , Proc. Internat. Congress Math. Cambridge, I (1950) pp. 470
[a11] N.J. Young, "The irregularity of multiplication in group algebras" Quart. J. Math. Oxford Ser. (2) , 24 (1973) pp. 59–62
[a12] N.J. Young, "Semigroup algebras having regular multiplication" Studia Math. , 47 (1973) pp. 191–196
[a13] N.J. Young, "Periodicity of functionals and representations of normed algebras on reflexive spaces" Proc. Edinburgh Math. Soc. (2) , 20 : 2 (1976–77) pp. 99–120
How to Cite This Entry:
Arens regularity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Arens_regularity&oldid=13185
This article was adapted from an original article by T.W. Palmer (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article