Introverted subspaces of the duals of measure algebras.

*(English)*Zbl 1398.43001Summary: Let \(\mathcal{G} \) be a locally compact group. In continuation of our studies on the first and second duals of measure algebras by the use of the theory of generalized functions, here we study the \(\mathrm{C}^*\)-subalgebra \(GL_0(\mathcal{G})\) of \(GL(\mathcal{G})\) as an introverted subspace of \(M(\mathcal{G})^*\). In the case where \(\mathcal{G} \) is non-compact, we show that any topological left invariant mean on \(GL(\mathcal{G})\) lies in \(GL_0(\mathcal{G})^\bot\). We then endow \(GL_0(\mathcal{G})^*\) with an Arens-type product, which contains \(M(\mathcal{G})\) as a closed subalgebra and \(M_a(\mathcal{G})\) as a closed ideal, which is a solid set with respect to absolute continuity in \(GL_0(\mathcal{G})^*\). Among other things, we prove that \(\mathcal{G}\) is compact if and only if \(GL_0(\mathcal{G})^*\) has a non-zero left (weakly) completely continuous element.

##### MSC:

43A10 | Measure algebras on groups, semigroups, etc. |

43A15 | \(L^p\)-spaces and other function spaces on groups, semigroups, etc. |

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

47B07 | Linear operators defined by compactness properties |

##### Keywords:

measure algebra; generalized functions vanishing at infinity; introverted subspace; topological invariant mean; completely continuous element##### References:

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