Let G be a topological group that need not be abelian, and denote by C(G) the algebra of continuous, complex valued functions on G. We characterize the continuous solutions <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$f,g\in C(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>∈</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of the sine addition law <Equation ID="Equ13"> <EquationSource Format="TEX">$\begin{aligned} f(x\tau (y))=f(x)g(y)+g(x)f(y),\ x,y\in G, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mi>τ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="4pt" /> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>G</mi> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>in which <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\tau $</EquationSource> <EquationSource Format="MATHML"><math> <mi>τ</mi> </math></EquationSource> </InlineEquation> is a continuous involution on G, meaning that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\tau (xy)=\tau (y)\tau (x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>τ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>τ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mi>τ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\tau (\tau (x))=x$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>τ</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$x,y\in G.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>G</mi> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation>

The sine addition law with an involution on groups

期刊论文
发表日期：

2026年2月19日

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Mohamed Talla,
Driss Zeglami,
Mohamed El Fatini

摘要

Let G be a topological group that need not be abelian, and denote by C(G) the algebra of continuous, complex valued functions on G. We characterize the continuous solutions $f,g\in C(G)$ $f, g \in C (G)$ of the sine addition law $\begin{aligned} f(x\tau (y))=f(x)g(y)+g(x)f(y),\ x,y\in G, \end{aligned}$ $\begin{matrix} f (x τ (y)) = f (x) g (y) + g (x) f (y), x, y \in G, \end{matrix}$ in which $\tau $ $τ$ is a continuous involution on G, meaning that $\tau (xy)=\tau (y)\tau (x)$ $τ (x y) = τ (y) τ (x)$ and $\tau (\tau (x))=x$ $τ (τ (x)) = x$ for all $x,y\in G.$ $x, y \in G .$