Let X and Y be locally compact Hausdorff spaces. We denote by <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$C_0^+(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <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> the cone of all non-negative real-valued continuous functions on X vanishing at infinity. In this paper, we consider a 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> satisfying the following two norm conditions for all <InlineEquation ID="IEq3"> <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>: <Equation ID="Equ7"> <EquationSource Format="TEX">$ \Vert T(f+g) \Vert = \Vert T(f)+T(g) \Vert ,\qquad \Vert T(f \cdot g) \Vert = \Vert T(f) \cdot T(g) \Vert . $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mo stretchy="false">‖</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo>+</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">‖</mo> <mo>=</mo> <mo stretchy="false">‖</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">‖</mo> <mo>,</mo> <mspace width="2em" /> <mo stretchy="false">‖</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo>·</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">‖</mo> <mo>=</mo> <mo stretchy="false">‖</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>·</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">‖</mo> <mo>.</mo> </mrow> </math></EquationSource> </Equation>The main result of this paper is that such a map T is a composition operator of the form <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$T(f) = f \circ \tau $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo>∘</mo> <mi>τ</mi> </mrow> </math></EquationSource> </InlineEquation>, induced by a homeomorphism <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\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>. As a direct consequence of our main theorem, we obtain a positive cone version of the Gelfand–Kolmogoroff theorem.

Additive and multiplicative maps in norm on the positive cone of continuous function algebras

期刊论文
发表日期：

2026年7月30日

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Takeshi Miura,
Natsumi Shibata

摘要

Let X and Y be locally compact Hausdorff spaces. We denote by $C_0^+(X)$ $C_{0}^{+} (X)$ the cone of all non-negative real-valued continuous functions on X vanishing at infinity. In this paper, we consider a bijection $T:C_0^+(X) \rightarrow C_0^+(Y)$ $T : C_{0}^{+} (X) \to C_{0}^{+} (Y)$ satisfying the following two norm conditions for all $f, g \in C_0^+(X)$ $f, g \in C_{0}^{+} (X)$ : $ \Vert T(f+g) \Vert = \Vert T(f)+T(g) \Vert ,\qquad \Vert T(f \cdot g) \Vert = \Vert T(f) \cdot T(g) \Vert . $ $‖ T (f + g) ‖ = ‖ T (f) + T (g) ‖, ‖ T (f \cdot g) ‖ = ‖ T (f) \cdot T (g) ‖ .$ The main result of this paper is that such a map T is a composition operator of the form $T(f) = f \circ \tau $ $T (f) = f \circ τ$ , induced by a homeomorphism $\tau :Y \rightarrow X$ $τ : Y \to X$ . As a direct consequence of our main theorem, we obtain a positive cone version of the Gelfand–Kolmogoroff theorem.