Second duals of measure algebras
Volume 481 / 2011
Abstract
Let $G$ be a locally compact group. We shall study the Banach algebras which are the group algebra $L^1(G)$ and the measure algebra $M(G)$ on $G$, concentrating on their second dual algebras. As a preliminary we shall study the second dual $C_0(\Omega)''$ of the $C^*$-algebra $C_0(\Omega)$ for a locally compact space $\Omega$, recognizing this space as $C(\newcommand{\WO}{\widetilde\Omega}\WO)$, where $\newcommand{\WO}{\widetilde\Omega}\WO$ is the hyper-Stonean envelope of $\Omega$.
We shall study the $C^*$-algebra $B^{b}(\Omega)$ of bounded Borel functions on $\Omega$, and we shall determine the exact cardinality of a variety of subsets of $\newcommand{\WO}{\widetilde\Omega}\WO$ that are associated with $B^{b}(\Omega)$.
We shall identify the second duals of the measure algebra $(M(G), \mathbin{\star})$ and the group algebra $(L^1(G), \mathbin{\star})$ as the Banach algebras $(M(\newcommand{\WG}{\widetilde{G}}\WG),\newcommand {\B}{\mathbin{\Box}} \B)$ and $(M(\Phi),\newcommand {\B}{\mathbin{\Box}} \B)$, respectively, where $\newcommand {\B}{\mathbin{\Box}}\B$ denotes the first Arens product and $\newcommand{\WG}{\widetilde{G}}\WG$ and $\Phi$ are certain compact spaces, and we shall then describe many of the properties of these two algebras. In particular, we shall show that the hyper-Stonean envelope $\newcommand{\WG}{\widetilde{G}}\WG$ determines the locally compact group $G$. We shall also show that $(\newcommand{\WG}{\widetilde{G}}\WG, \newcommand {\B}{\mathbin{\Box}}\B)$ is a semigroup if and only if $G$ is discrete, and we shall discuss in considerable detail the product of point masses in $M(\newcommand{\WG}{\widetilde{G}}\WG)$. Some important special cases will be considered.
We shall show that the spectrum of the $C^*$-algebra $L^\infty(G)$ is determining for the left topological centre of $L^1(G)''$, and we shall discuss the topological centre of the algebra $(M(G)'',\newcommand {\B}{\mathbin{\Box}} \B)$.
