We investigate quantum group generalizations of various density results from Fourier analysis on compact groups. In particular, we establish the density of characters in the space of fixed points of the conjugation action on ${{L}^{2}}(\mathbb{G})$ and use this result to show the $\text{wea}{{\text{k}}^{\star }}$ density and normal density of characters in $Z{{L}^{\infty }}(\mathbb{G})$ and $ZC(\mathbb{G})$, respectively. As a corollary, we partially answer an open question of Woronowicz. At the level of ${{L}^{1}}(\mathbb{G})$, we show that the center $~z({{L}^{1}}(\mathbb{G}))$ is precisely the closed linear span of the quantum characters for a large class of compact quantum groups, including arbitrary compact Kac algebras. In the latter setting, we show, in addition, that $~z({{L}^{1}}(\mathbb{G}))$ is a completely complemented $~z({{L}^{1}}(\mathbb{G}))$-submodule of ${{L}^{2}}(\mathbb{G})$.

For a locally compact group $G$, let $A(G)$ be its Fourier algebra, let ${{M}_{cb}}A(G)$ denote the completely bounded multipliers of $A(G)$, and let ${{A}_{Mcb}}\,(G)$ stand for the closure of $A(G)$ in ${{M}_{cb}}A(G)$. We characterize the norm one idempotents in ${{M}_{cb}}A(G)$: the indicator function of a set $E\,\subset \,G$ is a norm one idempotent in ${{M}_{cb}}A(G)$ if and only if $E$ is a coset of an open subgroup of $G$. As applications, we describe the closed ideals of ${{A}_{Mcb}}\,(G)$ with an approximate identity bounded by 1, and we characterize those $G$ for which ${{A}_{Mcb}}\,(G)$ is 1-amenable in the sense of B. E. Johnson. (We can even slightly relax the norm bounds.)

Let $G$ be a locally compact group, and let ${{A}_{\text{cb}}}(G)$ 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 ${{C}^{*}}(G)$ is residually finite-dimensional, we show that ${{A}_{\text{cb}}}(G)$ is operator amenable. In particular, ${{A}_{\text{cb}}}({{\mathbb{F}}_{2}})$ is operator amenable even though ${{\mathbb{F}}_{2}}$, the free group in two generators, is not an amenable group. Moreover, we show that if $G$ is a discrete group such that ${{A}_{\text{cb}}}(G)$ 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 $\text{cb}$-multiplier norm.

Let φ be a continuous nonzero homomorphism of the convolution algebra L1loc(R+) and also the unique extension of this homomorphism to Mloc(R+). We show that the map φis continuous in the weak* and strong opertor topologies on Mloc, considered as the dual space of Cc(R+) and as the multiplier algebra of L1loc. Analogous results are proved for homomorphism from L1 [0, a) to L1 [0, b). For each convolution algebra L1 (ω1), φ restricts to a continuous homomorphism from some L1 (ω1) to some L1 (ω2), and, for each sufficiently large L1 (ω2), φ restricts to a continuous homomorphism from some L1 (ω1) to L1 (ω2). We also determine which continuous homomorphisms between weighted convolution algebras extend to homomorphisms of L1loc. We also prove results on convergent nets, continuous semigroups, and bounded sets in Mloc that we need in our study of homomorphisms.

The $\ell^{1}$-convolution algebra of a semilattice is known to have trivial cohomology in degrees 1, 2 and 3 whenever the coefficient bimodule is symmetric. We extend this result to all cohomology groups of degree $\geq 1$ with symmetric coefficients. Our techniques prove a stronger splitting result, namely that the splitting can be made natural with respect to the underlying semilattice.

Thomas conjectured that there is no such thing as an almost prime element in a commutative radical Banach algebra. This paper shows that there is such an element. However, it still leaves open the question of whether there is an element that is almost prime, but not prime.

We prove a quantized version of a theorem by M. V. Sheĭnberg: A uniform algebra equipped with its canonical, i.e., minimal, operator space structure is operator amenable if and only if it is a commutative ${{C}^{*}}$-algebra.