# Arens regularity

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=45217
This article was adapted from an original article by T.W. Palmer (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article