We first show that for a -Banach function algebra A on a compact Hausdorff space X, any multiplicative function <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\varphi :\textrm{exp}(A) \rightarrow \mathbb {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mo>:</mo> <mtext>exp</mtext> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">C</mi> </mrow> </math></EquationSource> </InlineEquation> satisfying <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\varphi (f) \in f(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$f\in A$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <mi>A</mi> </mrow> </math></EquationSource> </InlineEquation>, is the restriction of an evaluation homomorphism. Then we show that if <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\varphi :C(X) \rightarrow \mathbb {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mo>:</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">C</mi> </mrow> </math></EquationSource> </InlineEquation> is a multiplicative function (not necessarily continuous) such that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\varphi (f) \in f(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$f\in C(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, then either <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\textrm{ker}(\varphi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>ker</mtext> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is a maximal ideal of C(X) or <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$1=f_1+ \cdots +f_n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>=</mo> <msub> <mi>f</mi> <mn>1</mn> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> for some <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$f_1,..., f_n\in \textrm{ker}(\varphi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>f</mi> <mi>n</mi> </msub> <mo>∈</mo> <mtext>ker</mtext> <mrow> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Meanwhile, <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation> is a character on C(X) in either of the cases that <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation> is continuous or <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$1\notin \textrm{span}(\textrm{ker}(\varphi ))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>∉</mo> <mtext>span</mtext> <mo stretchy="false">(</mo> <mtext>ker</mtext> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. In particular, this provides a short proof for the recent result concerning the linearity of continuous multiplicative spectral functions on commutative unital <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$C^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-algebras.

Linearity of Multiplicative Spectral Functions on C(X)

期刊论文
发表日期：

2026年4月11日

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Nahid Bayati,
Fereshteh Sady

摘要

We first show that for a *-Banach function algebra A on a compact Hausdorff space X, any multiplicative function $\varphi :\textrm{exp}(A) \rightarrow \mathbb {C}$ $φ : exp (A) \to C$ satisfying $\varphi (f) \in f(X)$ $φ (f) \in f (X)$ for all $f\in A$ $f \in A$ , is the restriction of an evaluation homomorphism. Then we show that if $\varphi :C(X) \rightarrow \mathbb {C}$ $φ : C (X) \to C$ is a multiplicative function (not necessarily continuous) such that $\varphi (f) \in f(X)$ $φ (f) \in f (X)$ for all $f\in C(X)$ $f \in C (X)$ , then either $\textrm{ker}(\varphi )$ $ker (φ)$ is a maximal ideal of C(X) or $1=f_1+ \cdots +f_n$ $1 = f_{1} + \dots + f_{n}$ for some $f_1,..., f_n\in \textrm{ker}(\varphi )$ $f_{1}, . . ., f_{n} \in ker (φ)$ . Meanwhile, $\varphi $ $φ$ is a character on C(X) in either of the cases that $\varphi $ $φ$ is continuous or $1\notin \textrm{span}(\textrm{ker}(\varphi ))$ $1 \notin span (ker (φ))$ . In particular, this provides a short proof for the recent result concerning the linearity of continuous multiplicative spectral functions on commutative unital $C^*$ $C^{*}$ -algebras.