# Arens regularity

A normed algebra 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 are identical. Since both Arens products extend the product on (relative to the natural embedding mapping ), 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 -algebra has a natural interpretation as the von Neumann algebra generated by the universal -representation of . Hence -algebras are always Arens regular.

It is easy to show that if is commutative under either Arens product, then 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) is Arens regular;

b) for each the adjoint of the left regular representation is weakly compact;

c) for each the adjoint of the right regular representation is weakly compact;

d) for any bounded sequences and in and any , the iterated limits

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) , (or ) is Arens regular if and only if 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) of a locally compact semi-group in which multiplication is at least singly continuous is Arens regular if and only if is. These are, in turn, equivalent to either:

there do not exist sequences and in such that the sets and are disjoint;

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

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

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

A weaker version of Arens regularity was introduced by M. Grosser [a4]. An approximately unital Banach algebra is said to be semi-regular if it satisfies for all mixed identities . (An element 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 |

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Arens regularity. T.W. Palmer (originator),

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