We study locally compact quantum groups $\mathbb{G}$ through the convolution algebras $L_1(\mathbb{G})$ and $(T(L_2(\mathbb{G})), \triangleright)$. We prove that the reduced quantum group $C^*$-algebra $C_0(\mathbb{G})$ can be recovered from the convolution $\triangleright$ by showing that the right $T(L_2(\mathbb{G}))$-module $\langle K(L_2(\mathbb{G}) \triangleright T(L_2(\mathbb{G}))\rangle$ is equal to $C_0(\mathbb{G})$. On the other hand, we show that the left $T(L_2(\mathbb{G}))$-module $\langle T(L_2(\mathbb{G}))\triangleright K(L_2(\mathbb{G})\rangle$ is isomorphic to the reduced crossed product $C_0(\widehat{\mathbb{G}}) \,_r\!\ltimes C_0(\mathbb{G})$, and hence is a much larger $C^*$-subalgebra of $B(L_2(\mathbb{G}))$. We establish a natural isomorphism between the completely bounded right multiplier algebras of $L_1(\mathbb{G})$ and $(T(L_2(\mathbb{G})), \triangleright)$, and settle two invariance problems associated with the representation theorem of Junge-Neufang-Ruan (2009). We characterize regularity and discreteness of the quantum group $\mathbb{G}$ in terms of continuity properties of the convolution $\triangleright$ on $T(L_2(\mathbb{G}))$. We prove that if $\mathbb{G}$ is semi-regular, then the space $\langle T(L_2(\mathbb{G}))\triangleright B(L_2(\mathbb{G}))\rangle$ of right $\mathbb{G}$-continuous operators on $L_2(\mathbb{G})$, which was introduced by Bekka (1990) for $L_{\infty}(G)$, is a unital $C^*$-subalgebra of $B(L_2(\mathbb{G}))$. In the representation framework formulated by Neufang-Ruan-Spronk (2008) and Junge-Neufang-Ruan, we show that the dual properties of compactness and discreteness can be characterized simultaneously via automatic normality of quantum group bimodule maps on $B(L_2(\mathbb{G}))$. We also characterize some commutation relations of completely bounded multipliers of $(T(L_2(\mathbb{G})), \triangleright)$ over $B(L_2(\mathbb{G}))$.

We show that the multiplier algebra of the Fourier algebra on a locally compact group $G$ can be isometrically represented on a direct sum on non-commutative $L^p$ spaces associated with the right von Neumann algebra of $G$. The resulting image is the idealiser of the image of the Fourier algebra. If these spaces are given their canonical operator space structure, then we get a completely isometric representation of the completely bounded multiplier algebra. We make a careful study of the non-commutative $L^p$ spaces we construct and show that they are completely isometric to those considered recently by Forrest, Lee, and Samei. We improve a result of theirs about module homomorphisms. We suggest a definition of a Figa-Talamanca-Herz algebra built out of these non-commutative $L^p$ spaces, say $A_p(\widehat G)$. It is shown that $A_2(\widehat G)$ is isometric to $L^1(G)$, generalising the abelian situation.

We investigate generalized amenability and biflatness properties of various (operator) Segal algebras in both the group algebra, $L^1(G)$, and the Fourier algebra, $A(G)$, of a locally compact group~$G$.

Let $G$ be a locally compact group, and let $A_{\cb}(G)$ denote the closure of $A(G)$, the Fourier algebra of $G$, in the space of completely bounded multipliers of $A(G)$. If $G$ is a weakly amenable, discrete group such that $\cstar(G)$ is residually finite-dimensional, we show that $A_{\cb}(G)$ is operator amenable. In particular, $A_{\cb}(\free_2)$ is operator amenable even though $\free_2$, the free group in two generators, is not an amenable group. Moreover, we show that if $G$ is a discrete group such that $A_{\cb}(G)$ is operator amenable, a closed ideal of $A(G)$ is weakly completely complemented in $A(G)$ if and only if it has an approximate identity bounded in the $\cb$-multiplier norm.

The decomposability number of a von Neumann algebra $\m$ (denoted by $\dec(\m)$) is the greatest cardinality of a family of pairwise orthogonal non-zero projections in $\m$. In this paper, we explore the close connection between $\dec(\m)$ and the cardinal level of the Mazur property for the predual $\m_*$ of $\m$, the study of which was initiated by the second author. Here, our main focus is on those von Neumann algebras whose preduals constitute such important Banach algebras on a locally compact group $G$ as the group algebra $\lone$, the Fourier algebra $A(G)$, the measure algebra $M(G)$, the algebra $\luc^*$, etc. We show that for any of these von Neumann algebras, say $\m$, the cardinal number $\dec(\m)$ and a certain cardinal level of the Mazur property of $\m_*$ are completely encoded in the underlying group structure. In fact, they can be expressed precisely by two dual cardinal invariants of $G$: the compact covering number $\kg$ of $G$ and the least cardinality $\bg$ of an open basis at the identity of $G$. We also present an application of the Mazur property of higher level to the topological centre problem for the Banach algebra $\ag^{**}$.

Let $\VN(G)$ be the von Neumann algebra of a locally compact group $G$. We denote by $\mu$ the initial ordinal with $\abs{\mu}$ equal to the smallest cardinality of an open basis at the unit of $G$ and $X= \{\alpha; \alpha < \mu \}$. We show that if $G$ is nondiscrete then there exist an isometric $*$-isomorphism $\kappa$ of $l^{\infty}(X)$ into $\VN(G)$ and a positive linear mapping $\pi$ of $\VN(G)$ onto $l^{\infty}(X)$ such that $\pi\circ\kappa = \id_{l^{\infty}(X)}$ and $\kappa$ and $\pi$ have certain additional properties. Let $\UCB (\hat{G})$ be the $C^{*}$-algebra generated by operators in $\VN(G)$ with compact support and $F(\hat{G})$ the space of all $T \in \VN(G)$ such that all topologically invariant means on $\VN(G)$ attain the same value at $T$. The construction of the mapping $\pi$ leads to the conclusion that the quotient space $\UCB (\hat{G})/F(\hat{G})\cap \UCB(\hat{G})$ has $l^{\infty}(X)$ as a continuous linear image if $G$ is nondiscrete. When $G$ is further assumed to be non-metrizable, it is shown that $\UCB(\hat{G})/F (\hat{G})\cap \UCB(\hat{G})$ contains a linear isomorphic copy of $l^{\infty}(X)$. Similar results are also obtained for other quotient spaces.