<p>Let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathfrak {B}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">B</mi> </math></EquationSource> </InlineEquation> be unital <InlineEquation ID="IEq5"> <EquationSource Format="TEX"></EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow /> <mo>∗</mo> </mrow> </math></EquationSource> </InlineEquation>-algebras such that <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation> has a nontrivial projection. In this present article, we demonstrate, under certain restrictions that if a bijective map <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\Omega :\mathcal {A}\rightarrow \mathfrak {B}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>:</mo> <mi mathvariant="script">A</mi> <mo stretchy="false">→</mo> <mi mathvariant="fraktur">B</mi> </mrow> </math></EquationSource> </InlineEquation> satisfies <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\Omega ([E^{}\circ F, G]^)=[\Omega (E)^{}\circ \Omega (F), \Omega (G)]^$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mrow> <mo stretchy="false">(</mo> <mmultiscripts> <mrow> <mo stretchy="false">[</mo> <mmultiscripts> <mi>E</mi> <mrow /> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mo>∘</mo> <mi>F</mi> <mo>,</mo> <mi>G</mi> <mo stretchy="false">]</mo> </mrow> <mrow /> <mo>∗</mo> </mmultiscripts> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mmultiscripts> <mrow> <mo stretchy="false">[</mo> <mi mathvariant="normal">Ω</mi> <mmultiscripts> <mrow> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mrow> <mrow /> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mo>∘</mo> <mi mathvariant="normal">Ω</mi> <mrow> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mi mathvariant="normal">Ω</mi> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">]</mo> </mrow> <mrow /> <mo>∗</mo> </mmultiscripts> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$E, F, G \in \mathcal {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>E</mi> <mo>,</mo> <mi>F</mi> <mo>,</mo> <mi>G</mi> <mo>∈</mo> <mi mathvariant="script">A</mi> </mrow> </math></EquationSource> </InlineEquation>, then <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> is a <InlineEquation ID="IEq11"> <EquationSource Format="TEX"></EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow /> <mo>∗</mo> </mrow> </math></EquationSource> </InlineEquation>-preserving ring isomorphism. Also, this result is applied to factor von Neumann algebras.</p>

Nonlinear maps preserving the mixed type products on $*$-algebras

期刊论文
发表日期：

2025年5月2日

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Mohammad Aslam Siddeeque,
Mohammad Shane Alam,
Raof Ahmad Bhat

摘要

Let $\mathcal {A}$ $A$ and $\mathfrak {B}$ $B$ be unital $*$ $*$ -algebras such that $\mathcal {A}$ $A$ has a nontrivial projection. In this present article, we demonstrate, under certain restrictions that if a bijective map $\Omega :\mathcal {A}\rightarrow \mathfrak {B}$ $Ω : A \to B$ satisfies $\Omega ([E^{*}\circ F, G]^*)=[\Omega (E)^{*}\circ \Omega (F), \Omega (G)]^*$ $Ω ({[E^{*} \circ F, G]}^{*}) = {[Ω {(E)}^{*} \circ Ω (F), Ω (G)]}^{*}$ for all $E, F, G \in \mathcal {A}$ $E, F, G \in A$ , then $\Omega $ $Ω$ is a $*$ $*$ -preserving ring isomorphism. Also, this result is applied to factor von Neumann algebras.

Nonlinear maps preserving the mixed type products on \(*\)-algebras

摘要