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Let A be a semisimple commutative regular tauberian Banach algebra with spectrum ${\Sigma }_A$. In this paper, we study the norm spectra of elements of $\overline{span}{\Sigma }_A$ and present some applications. In particular, we characterize the discreteness of ${\Sigma }_A$ in terms of norm spectra. The algebra A is said to have property (S) if, for all $\phi \in \overline{}{\Sigma }_{A}0$, φ has a nonempty norm spectrum. For a locally compact group G, let ${ℳ}_2^d\left(Ĝ\right)$ denote the C*-algebra generated by left translation operators on $L^2\left(G\right)$ and $G_d$ denote the discrete group G. We prove that the Fourier...

Let G be a non-discrete locally compact group, A(G) the Fourier algebra of G, VN(G) the von Neumann algebra generated by the left regular representation of G which is identified with A(G)*, and WAP(Ĝ) the space of all weakly almost periodic functionals on A(G). We show that there exists a directed family ℋ of open subgroups of G such that: (1) for each H ∈ ℋ, A(H) is extremely non-Arens regular; (2) $VN\left(G\right)={\bigcup }_{H\in \mathscr{H}}VN\left(H\right)$ and $VN\left(G\right)/WAP\left(Ĝ\right)={\bigcup }_{H\in \mathscr{H}}[VN\left(H\right)/WAP\left(Ĥ\right)]$; (3) $A\left(G\right)={\bigcup }_{H\in \mathscr{H}}A\left(H\right)$ and it is a WAP-strong inductive union in the sense that the unions in (2) are strongly...

We study locally compact quantum groups and their module maps through a general Banach algebra approach. As applications, we obtain various characterizations of compactness and discreteness, which in particular generalize a result by Lau (1978) and recover another one by Runde (2008). Properties of module maps on $L_{\infty }\left(\right)$ are used to characterize strong Arens irregularity of L₁() and are linked to commutation relations over with several double commutant theorems established. We prove the quantum group...