<p>Conic quasi-linear maps are nonlinear operators from <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> to a normed linear space <i>E</i> which preserve nonnegative linear combinations on positive cones generated by single functions; quasi-linear maps are linear on singly generated subalgebras. While nonlinear, for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$E = C_b(Y)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>E</mi> <mo>=</mo> <msub> <mi>C</mi> <mi>b</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> a quasi-linear map is bounded iff it is continuous. <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$E = {\mathbb {R}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>E</mi> <mo>=</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> gives quasi-integrals, which correspond to (deficient) topological measures—nonsubadditive set functions generalizing measures. Like image measures <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mu \circ u^{-1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>μ</mi> <mo>∘</mo> <msup> <mi>u</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>, (d-) image transformations move (deficient) topological measures from one space to another, generalizing <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$u^{-1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>u</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>. We give criteria for a (d-) image transformation to be <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$u^{-1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>u</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> for some proper continuous function. We study the interrelationships between (conic) quasi-linear maps, quasi-integrals, (deficient) topological measures and (d-) image transformations when <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$E = C_0(Y), X, Y$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>E</mi> <mo>=</mo> <msub> <mi>C</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> </mrow> </math></EquationSource> </InlineEquation> are locally compact. (Conic) quasi-homomorphisms behave like homomorphisms on singly generated subalgebras or cones. We show by construction that (conic) quasi-homomorphisms are in 1-1 correspondence with (d-) image transformations and with certain continuous proper functions. We give criteria for a (conic) quasi-linear map to be a (conic) quasi-homomorphism, and for the latter to be an algebra homomorphism. We show that one may approach a well-known result from Gelfand duality theory from the perspective of quasi-homomorphisms and image transformations. Any conic quasi-linear map or quasi-linear map is a composition of an algebra homomorphism with the basic quasi-linear map, and we give criteria for the latter to be linear. We study the adjoints of (d-) image transformations and (conic) quasi-linear maps; for (conic) quasi-homomorphisms they give Markov–Feller operators with nonlinear duals.</p>

Quasi-linear maps and image transformations

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2026年7月31日

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S. V. Butler

摘要

Conic quasi-linear maps are nonlinear operators from $C_0(X)$ $C_{0} (X)$ to a normed linear space E which preserve nonnegative linear combinations on positive cones generated by single functions; quasi-linear maps are linear on singly generated subalgebras. While nonlinear, for $E = C_b(Y)$ $E = C_{b} (Y)$ a quasi-linear map is bounded iff it is continuous. $E = {\mathbb {R}}$ $E = R$ gives quasi-integrals, which correspond to (deficient) topological measures—nonsubadditive set functions generalizing measures. Like image measures $\mu \circ u^{-1}$ $μ \circ u^{- 1}$ , (d-) image transformations move (deficient) topological measures from one space to another, generalizing $u^{-1}$ $u^{- 1}$ . We give criteria for a (d-) image transformation to be $u^{-1}$ $u^{- 1}$ for some proper continuous function. We study the interrelationships between (conic) quasi-linear maps, quasi-integrals, (deficient) topological measures and (d-) image transformations when $E = C_0(Y), X, Y$ $E = C_{0} (Y), X, Y$ are locally compact. (Conic) quasi-homomorphisms behave like homomorphisms on singly generated subalgebras or cones. We show by construction that (conic) quasi-homomorphisms are in 1-1 correspondence with (d-) image transformations and with certain continuous proper functions. We give criteria for a (conic) quasi-linear map to be a (conic) quasi-homomorphism, and for the latter to be an algebra homomorphism. We show that one may approach a well-known result from Gelfand duality theory from the perspective of quasi-homomorphisms and image transformations. Any conic quasi-linear map or quasi-linear map is a composition of an algebra homomorphism with the basic quasi-linear map, and we give criteria for the latter to be linear. We study the adjoints of (d-) image transformations and (conic) quasi-linear maps; for (conic) quasi-homomorphisms they give Markov–Feller operators with nonlinear duals.