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Let \omega :\mathbb Z^2 \to [1,\infty) be a weight satisfying a certain technical property. Let f be a continuous function on the torus \mathbb T^2 such that \sum _{\mathbf {m}\in \mathbb Z^2}|\widehat f(\mathbf {m})|\omega (\mathbf {m})<\infty. If f is nowhere vanishing on \mathbb T^2, then there is a weight \nu on \mathbb Z^2 such that 1\leq \nu \leq \omega, \nu is nonconstant if and only if \omega is nonconstant and \sum _{\mathbf {m}\in \mathbb Z^2}\left |\widehat {(\frac {1}{f})}(\mathbf {m})\right |\nu (\mathbf {m})<\infty. If \varphi is holomorphic on a neighbourhood of the range of f, then there is weight \chi on \mathbb Z^2 such that 1\leq \chi \leq \omega, \chi is nonconstant if and only if \omega is nonconstant and \sum _{\mathbf {m}\in \mathbb Z^2}|\widehat {(\varphi \circ f)}(\mathbf {m})|\chi (\mathbf {m})<\infty. An analogue of Żelazko's theorem on p-th power convergence is also discussed. These two variables weighted analogues of classical theorems of Wiener, Lévy and Żelazko are proved exhibiting the complexity of weights in two variables. [ABSTRACT FROM AUTHOR]

Let $(G_1,\omega _1)$ and $(G_2,\omega _2)$ be weighted discrete groups and $0. We characterise biseparating bicontinuous algebra isomorphisms on the p-Banach algebra $\ell ^p(G_1,\omega _1)$. We also characterise bipositive and isometric algebra isomorphisms between the p-Banach algebras $\ell ^p(G_1,\omega _1)$ and $\ell ^p(G_2,\omega _2)$ and isometric algebra isomorphisms between $\ell ^p(S_1,\omega _1)$ and $\ell ^p(S_2,\omega _2)$ , where $(S_1,\omega _1)$ and $(S_2,\omega _2)$ are weighted discrete semigroups. [ABSTRACT FROM AUTHOR]

Let A and B be commutative Banach algebras. Then a multi- plicative linear functional θ on B induces a multiplication on the Cartesian product space A × B given by (a,b)(c,d) = (ac + θ(d)a + θ(b)c,bd) for all (a,b),(c,d) ∊ A × B. We show that this Lau product is stable with respect to the spectral properties like the unique uniform norm property, the spectral extension property, the multiplicative Hahn{Banach property, and the unique semisimple norm property under certain conditions on θ. [ABSTRACT FROM AUTHOR]