The equation $$\pmb {f(xy) = f(x)h(y) + g(x)f(y)}$$ and representations on $$\pmb {\mathbb {C}^2}$$,Aequationes Mathematicae

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The equation $$\pmb {f(xy) = f(x)h(y) + g(x)f(y)}$$ and representations on $$\pmb {\mathbb {C}^2}$$
Aequationes Mathematicae ( IF 0.8 ) Pub Date : 2023-11-16 , DOI: 10.1007/s00010-023-01014-4
Henrik Stetkær

Let G be a topological group, and let C(G) denote the algebra of continuous, complex valued functions on G. We find the solutions $f,g,h \in C(G)$ of the Levi-Civita equation

$$\begin{aligned} f(xy) = f(x)h(y) + g(x)f(y), \ x,y \in G, \end{aligned}$$

which is an extension of the sine addition law. Representations of G on $\mathbb {C}^2$ play an important role. As a corollary we get the solutions $f,g \in C(G)$ of the sine subtraction law $f(xy^*) = f(x)g(y) - g(x)f(y)$, $x,y \in G$, in which $x \mapsto x^*$ is a continuous involution, meaning that $(xy)^* = y^*x^*$ and $x^{**} = x$ for all $x,y \in G$.

更新日期：2023-11-18

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