The spine of a Fourier-Stieltjes algebra. (English) Zbl 1113.43004
In [J. Math. Soc. Japan 23, 278–294 (1971; Zbl 0212.15001)] J. Inoue defined, for a locally compact abelian group \(G\), a certain subalgebra \(L^\ast(G)\) of \(M(G)\) that was later dubbed the spine of \(M(G)\) by J. L. Taylor.
In the (very interesting) paper under review, the authors extend Inoue’s construction to general – not necessarily abelian – locally compact groups \(G\) by adopting a dual point of view and define the spine \(A^\ast(G)\) as a closed subalgebra of \(B(G)\), Eymard’s Fourier–Stieltjes algebra. We briefly outline their construction.
Let \({\mathcal T}(G)\) denote the collection of all so-called locally precompact group topologies on \(G\), i.e., each \(\tau \in {\mathcal T}(G)\) is a Hausdorff topology coarser than the given one such that the completion \(G_\tau\) of \(G\) with respect to the left (or, equivalently, right) uniformity generated by \(\tau\) is a locally compact group. It is straightforward that \(B(G)\) contains a canonical image \(A_\tau(G)\) of \(A(G_\tau)\) for each \(\tau \in {\mathcal T}(G)\). The authors define the spine of \(B(G)\) as \[ A^\ast(G) = \overline{\sum_{\tau \in {\mathcal T}(G)} A_\tau(G)}. \]
Let \(G^{ap}\) denote the almost periodic compactification of \(G\), and let \(\tau_{ap} \in {\mathcal T}(G)\) denote the topology arising from the canonical map from \(G\) into \(G^{ap}\). Those \(\tau \in {\mathcal T}(G)\) that are finer than \(\tau_{ap}\) are called non-quotient topologies; the collection of all non-quotient topologies is denoted by \({\mathcal T}_{nq}(G)\). Somewhat surprisingly, \({\mathcal T}_{nq}(G)\) is a semilattice, and \(A^\ast(G)\) is a graded Banach algebra over it.
The authors study \(A^\ast(G)\) in great detail. For instance, they describe its character space, and study the completely bounded homomorphisms from \(A^\ast(G)\) into \(B(H)\) where \(H\) is another locally compact group (this builds on earlier work of theirs; see [M. Ilie, M. and N. Spronk, J. Funct. Anal. 225, 480–499 (2005; Zbl 1077.43004)]). Finally, they give concrete descriptions of \(A^\ast(G)\) for a number of specific locally compact groups \(G\).
In the (very interesting) paper under review, the authors extend Inoue’s construction to general – not necessarily abelian – locally compact groups \(G\) by adopting a dual point of view and define the spine \(A^\ast(G)\) as a closed subalgebra of \(B(G)\), Eymard’s Fourier–Stieltjes algebra. We briefly outline their construction.
Let \({\mathcal T}(G)\) denote the collection of all so-called locally precompact group topologies on \(G\), i.e., each \(\tau \in {\mathcal T}(G)\) is a Hausdorff topology coarser than the given one such that the completion \(G_\tau\) of \(G\) with respect to the left (or, equivalently, right) uniformity generated by \(\tau\) is a locally compact group. It is straightforward that \(B(G)\) contains a canonical image \(A_\tau(G)\) of \(A(G_\tau)\) for each \(\tau \in {\mathcal T}(G)\). The authors define the spine of \(B(G)\) as \[ A^\ast(G) = \overline{\sum_{\tau \in {\mathcal T}(G)} A_\tau(G)}. \]
Let \(G^{ap}\) denote the almost periodic compactification of \(G\), and let \(\tau_{ap} \in {\mathcal T}(G)\) denote the topology arising from the canonical map from \(G\) into \(G^{ap}\). Those \(\tau \in {\mathcal T}(G)\) that are finer than \(\tau_{ap}\) are called non-quotient topologies; the collection of all non-quotient topologies is denoted by \({\mathcal T}_{nq}(G)\). Somewhat surprisingly, \({\mathcal T}_{nq}(G)\) is a semilattice, and \(A^\ast(G)\) is a graded Banach algebra over it.
The authors study \(A^\ast(G)\) in great detail. For instance, they describe its character space, and study the completely bounded homomorphisms from \(A^\ast(G)\) into \(B(H)\) where \(H\) is another locally compact group (this builds on earlier work of theirs; see [M. Ilie, M. and N. Spronk, J. Funct. Anal. 225, 480–499 (2005; Zbl 1077.43004)]). Finally, they give concrete descriptions of \(A^\ast(G)\) for a number of specific locally compact groups \(G\).
Reviewer: Volker Runde (Edmonton)
MSC:
| 43A30 | Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc. |
| 22D05 | General properties and structure of locally compact groups |
| 43A60 | Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions |
| 47L50 | Dual spaces of operator algebras |
| 46L07 | Operator spaces and completely bounded maps |
| 22B05 | General properties and structure of LCA groups |
Keywords:
Fourier–Stieltjes algebra; Fourier algebra; spine; locally precompact group topologies; non-quotient topologiesReferences:
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