##
**The second duals of Beurling algebras.**
*(English)*
Zbl 1075.43003

Mem. Am. Math. Soc. 836, 191 p. (2005).

Let \(A\) be a Banach algebra. There are two canonical ways to turn the second dual \(A''\) into a Banach algebra such that the product of \(A\) is extended: the first Arens product – denoted by \(\square\) in this memoir – and the second Arens product \(\diamondsuit\). In \((A'',\square)\) multiplication with a fixed element from the right is \(\sigma(A'',A')\)-continuous, and in \((A'',\diamondsuit)\) the same is true for multiplication from the left. If \((A'',\square) = (A'',\diamondsuit)\), the algebra \(A\) is called Arens regular. For instance, \(C^\ast\)-algebras are Arens regular [P. Civin and B. Yood, Pac. J. Math. 11, 847–870 (1961; Zbl 0119.10903)] as is \({\mathcal K}(E)\), the algebra of all compact operators on a Banach space \(E\), if \(E\) is reflexive [N. J. Young, Q. J. Math., Oxf. II. Ser. 24, 59–62 (1972; Zbl 0252.43009)]. The first topological center of \(A''\) is defined as
\[
\mathfrak{Z}_t^{(1)}(A'') := \{ \Phi \in A'' : \text{left multiplication with \(\Phi\) in \((A'',\square)\) is \(\sigma(A'',A')\)-continuous} \};
\]
the second topological center \(\mathfrak{Z}_t^{(2)}(A'')\) is defined similarly (with respect to right multiplication in \((A'',\diamondsuit)\)). One has \(A \subset \mathfrak{Z}_t^{(j)}(A'') \subset A''\) for \(j=1,2\), where both inclusions can be proper. Clearly, \(A\) is Arens regular if and only if \(\mathfrak{Z}_t^{(1)}(A'') = \mathfrak{Z}_t^{(2)}(A'') = A''\). The authors call \(A\) strongly Arens irregular if \(\mathfrak{Z}_t^{(1)}(A'') = \mathfrak{Z}_t^{(2)}(A'') = A\); for instance, group algebras \(L^1(G)\) for an infinite, locally compact group are strongly Arens irregular.

The memoir under review undertakes a systematic study of the Banach algebras \((A'',\square)\) and \((A'',\diamondsuit)\) as well as of \(\mathfrak{Z}_t^{(1)}(A'') = \mathfrak{Z}_t^{(1)}(A'')\). Special attention is given to the case where \(A\) is a Beurling algebra, i.e., a weighted group algebra. In general, \((A'',\square)\) and \((A'',\diamondsuit)\) can display surprisingly different behavior.

The memoir – especially its first seven chapters – is to some extent expository, which makes it quite accessible to the reader: the authors review and sometimes recast a number of known results. For example, they give a rather straightforward proof of M. Daws’ recent result that \({\mathcal B}(E)\), the Banach algebra of all bounded operators on a Banach space \(E\), is Arens regular if \(E\) is superreflexive [M. Daws, Bull. Lond. Math. Soc. 36, No. 4, 493–503 (2004; Zbl 1066.47073)].

For \(A = {\mathcal K}(E)\), where \(E\) is not reflexive, and \(E'\) has both the approximation property and the Radon-Nikodým property, the authors compute both \(\mathfrak{Z}^{(1)}(A'')\) and \(\mathfrak{Z}^{(2)}(A'')\) as well as their intersection. In this case, \(A \subsetneq \mathfrak{Z}^{(j)}(A'') \subsetneq A''\) holds for \(j=1,2\), and \(\mathfrak{Z}^{(1)}(A'') \cap \mathfrak{Z}^{(2)}(A'')\) may or may not equal \(A\). Also, \((A'',\square) (\cong {\mathcal B}(E''))\) is semisimple whereas \((A'', \diamondsuit)\) isn’t.

Chapters 7 to 13 are devoted to the study of Beurling algebras \(L^1(G,\omega)\), where \(G\) is a locally compact group and \(\omega\) a suitable weight. Special emphasis is devoted to the case where \(G\) is discrete and abelian or the free group in two generators. The phenomena that can occur in this context are manifold. For instance, let \(\alpha \geq 0\), define a weight \(\omega_\alpha\) on \(\mathbb Z\) by letting \(\omega_\alpha(n) := (1+| n| )^\alpha\) for \(n \in \mathbb Z\); it has long been known that \(\ell^1(\mathbb{Z},\omega_\alpha)\) is Arens regular if and only if \(\alpha > 0\). On the other hand, the authors construct a weight, which is increasing on \(\mathbb{Z}^+\), such that \(\ell^1(\mathbb{Z},\omega) \subsetneq \mathfrak{Z}^{(1)}_t(\ell^1(\mathbb{Z},\omega)'') \subsetneq \ell^1(\mathbb{Z},\omega)''\) and such that the Jacobson radical of \((\ell^1(\mathbb{Z},\omega)'',\square)\) is nilpotent of index three. Another example of a weight \(\omega\) is symmetric, unbounded and such that \(\ell^1(\mathbb{Z},\omega)\) is strongly Arens irregular; the authors suspect, but fail to prove, that \((\ell^1(\mathbb{Z},\omega)'',\square)\) is semisimple for this omega.

The memoir concludes with a list of open problems. One of them is whether there is a weight \(\omega\) on \(\mathbb Z\) such that \((\ell^1(\mathbb{Z},\omega)'',\square)\) is semisimple.

The memoir under review undertakes a systematic study of the Banach algebras \((A'',\square)\) and \((A'',\diamondsuit)\) as well as of \(\mathfrak{Z}_t^{(1)}(A'') = \mathfrak{Z}_t^{(1)}(A'')\). Special attention is given to the case where \(A\) is a Beurling algebra, i.e., a weighted group algebra. In general, \((A'',\square)\) and \((A'',\diamondsuit)\) can display surprisingly different behavior.

The memoir – especially its first seven chapters – is to some extent expository, which makes it quite accessible to the reader: the authors review and sometimes recast a number of known results. For example, they give a rather straightforward proof of M. Daws’ recent result that \({\mathcal B}(E)\), the Banach algebra of all bounded operators on a Banach space \(E\), is Arens regular if \(E\) is superreflexive [M. Daws, Bull. Lond. Math. Soc. 36, No. 4, 493–503 (2004; Zbl 1066.47073)].

For \(A = {\mathcal K}(E)\), where \(E\) is not reflexive, and \(E'\) has both the approximation property and the Radon-Nikodým property, the authors compute both \(\mathfrak{Z}^{(1)}(A'')\) and \(\mathfrak{Z}^{(2)}(A'')\) as well as their intersection. In this case, \(A \subsetneq \mathfrak{Z}^{(j)}(A'') \subsetneq A''\) holds for \(j=1,2\), and \(\mathfrak{Z}^{(1)}(A'') \cap \mathfrak{Z}^{(2)}(A'')\) may or may not equal \(A\). Also, \((A'',\square) (\cong {\mathcal B}(E''))\) is semisimple whereas \((A'', \diamondsuit)\) isn’t.

Chapters 7 to 13 are devoted to the study of Beurling algebras \(L^1(G,\omega)\), where \(G\) is a locally compact group and \(\omega\) a suitable weight. Special emphasis is devoted to the case where \(G\) is discrete and abelian or the free group in two generators. The phenomena that can occur in this context are manifold. For instance, let \(\alpha \geq 0\), define a weight \(\omega_\alpha\) on \(\mathbb Z\) by letting \(\omega_\alpha(n) := (1+| n| )^\alpha\) for \(n \in \mathbb Z\); it has long been known that \(\ell^1(\mathbb{Z},\omega_\alpha)\) is Arens regular if and only if \(\alpha > 0\). On the other hand, the authors construct a weight, which is increasing on \(\mathbb{Z}^+\), such that \(\ell^1(\mathbb{Z},\omega) \subsetneq \mathfrak{Z}^{(1)}_t(\ell^1(\mathbb{Z},\omega)'') \subsetneq \ell^1(\mathbb{Z},\omega)''\) and such that the Jacobson radical of \((\ell^1(\mathbb{Z},\omega)'',\square)\) is nilpotent of index three. Another example of a weight \(\omega\) is symmetric, unbounded and such that \(\ell^1(\mathbb{Z},\omega)\) is strongly Arens irregular; the authors suspect, but fail to prove, that \((\ell^1(\mathbb{Z},\omega)'',\square)\) is semisimple for this omega.

The memoir concludes with a list of open problems. One of them is whether there is a weight \(\omega\) on \(\mathbb Z\) such that \((\ell^1(\mathbb{Z},\omega)'',\square)\) is semisimple.

Reviewer: Volker Runde (Edmonton)

### MSC:

43A20 | \(L^1\)-algebras on groups, semigroups, etc. |

46H05 | General theory of topological algebras |

22D15 | Group algebras of locally compact groups |

46H25 | Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) |

47B47 | Commutators, derivations, elementary operators, etc. |

47L10 | Algebras of operators on Banach spaces and other topological linear spaces |

46J10 | Banach algebras of continuous functions, function algebras |