<p>We prove that the faithful and the uniqueness of norm properties are stable in different product algebras, such as direct-sum product algebras, convolution product algebras, and the module product algebras. Furthermore, we show that these properties are not stable in null product algebras. Also, we provide a common sufficient condition, in terms of the algebra norm, for the co-dimension of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal{A}^2 = \operatorname {span} \{ ab : a,b \in \mathcal{A}\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="script">A</mi> </mrow> <mn>2</mn> </msup> <mo>=</mo> <mo>span</mo> <mrow> <mo stretchy="false">{</mo> <mi>a</mi> <mi>b</mi> <mo>:</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi mathvariant="script">A</mi> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> to be finite in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal{A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal{A}^{2} = \mathcal{A}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="script">A</mi> </mrow> <mn>2</mn> </msup> <mo>=</mo> <mi mathvariant="script">A</mi> </mrow> </math></EquationSource> </InlineEquation>(whenever <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal{A}^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">A</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> being dense in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal{A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation>, i.e., <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\overline{\mathcal{A}^2} = \mathcal{A})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover> <msup> <mrow> <mi mathvariant="script">A</mi> </mrow> <mn>2</mn> </msup> <mo>¯</mo> </mover> <mrow> <mo>=</mo> <mi mathvariant="script">A</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>

Uniqueness of norm and faithfulness of some product Banach algebras

期刊论文
发表日期：

2026年3月23日

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H. V. Dedania,
J. G. Patel

摘要

We prove that the faithful and the uniqueness of norm properties are stable in different product algebras, such as direct-sum product algebras, convolution product algebras, and the module product algebras. Furthermore, we show that these properties are not stable in null product algebras. Also, we provide a common sufficient condition, in terms of the algebra norm, for the co-dimension of $\mathcal{A}^2 = \operatorname {span} \{ ab : a,b \in \mathcal{A}\}$ $A^{2} = span {a b : a, b \in A}$ to be finite in $\mathcal{A}$ $A$ and $\mathcal{A}^{2} = \mathcal{A}$ $A^{2} = A$ (whenever $\mathcal{A}^2$ $A^{2}$ being dense in $\mathcal{A}$ $A$ , i.e., $\overline{\mathcal{A}^2} = \mathcal{A})$ $\bar{A^{2}} = A)$ .