We study bijections between the positive cones of spaces of continuous functions vanishing at infinity that satisfy a norm additive condition. Such maps arise naturally in the study of nonlinear functional equations and norm-preserving structures on function spaces. While in the compact (unital) case these maps can often be analyzed via linear extension techniques, the non-unital setting <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$C_0(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> requires a different approach due to the absence of a distinguished unit element. In this paper, we show that every bijection <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$T:C_0^+(X)\rightarrow C_0^+(Y)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>:</mo> <msubsup> <mi>C</mi> <mn>0</mn> <mo>+</mo> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">→</mo> <msubsup> <mi>C</mi> <mn>0</mn> <mo>+</mo> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> between the positive cones of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$C_0(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$C_0(Y)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> satisfying <Equation ID="Equ3"> <EquationSource Format="TEX">$ \Vert T(f+g)\Vert =\Vert T(f)+T(g)\Vert $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mo stretchy="false">‖</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo>+</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">‖</mo> <mo>=</mo> <mo stretchy="false">‖</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">‖</mo> </mrow> </math></EquationSource> </Equation>for all <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$f,g\in C_0^+(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>∈</mo> <msubsup> <mi>C</mi> <mn>0</mn> <mo>+</mo> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> admits a representation of the form <Equation ID="Equ4"> <EquationSource Format="TEX">$ T(f)(y)=h(y)f(\tau (y)), $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi>T</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\tau :Y\rightarrow X$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>τ</mi> <mo>:</mo> <mi>Y</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation> is a homeomorphism and h is a bounded continuous function from Y to <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$(0,\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. This yields a complete characterization of norm additive bijections on positive cones of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$C_0(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.

Norm additive maps between the positive cones of continuous function algebras

期刊论文
发表日期：

2026年6月20日

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Natsumi Shibata,
Takeshi Miura

摘要

We study bijections between the positive cones of spaces of continuous functions vanishing at infinity that satisfy a norm additive condition. Such maps arise naturally in the study of nonlinear functional equations and norm-preserving structures on function spaces. While in the compact (unital) case these maps can often be analyzed via linear extension techniques, the non-unital setting $C_0(X)$ $C_{0} (X)$ requires a different approach due to the absence of a distinguished unit element. In this paper, we show that every bijection $T:C_0^+(X)\rightarrow C_0^+(Y)$ $T : C_{0}^{+} (X) \to C_{0}^{+} (Y)$ between the positive cones of $C_0(X)$ $C_{0} (X)$ and $C_0(Y)$ $C_{0} (Y)$ satisfying $ \Vert T(f+g)\Vert =\Vert T(f)+T(g)\Vert $ $‖ T (f + g) ‖ = ‖ T (f) + T (g) ‖$ for all $f,g\in C_0^+(X)$ $f, g \in C_{0}^{+} (X)$ admits a representation of the form $ T(f)(y)=h(y)f(\tau (y)), $ $T (f) (y) = h (y) f (τ (y)),$ where $\tau :Y\rightarrow X$ $τ : Y \to X$ is a homeomorphism and h is a bounded continuous function from Y to $(0,\infty )$ $(0, \infty)$ . This yields a complete characterization of norm additive bijections on positive cones of $C_0(X)$ $C_{0} (X)$ .