<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {B}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">B</mi> </math></EquationSource> </InlineEquation> be a separable simple stable purely infinite C-algebra, and let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {M}(\mathcal {B})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">M</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">B</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be the multiplier algebra of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {B}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">B</mi> </math></EquationSource> </InlineEquation>. We find a multiplier algebra analog of a result of Brown, Pearcy and Salinas, proving that for all <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$X \in \mathcal {M}(\mathcal {B})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>∈</mo> <mi mathvariant="script">M</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">B</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, there exists a nilpotent operator <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$N \in \mathcal {M}(\mathcal {B})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>∈</mo> <mi mathvariant="script">M</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">B</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for which <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$X + N$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>+</mo> <mi>N</mi> </mrow> </math></EquationSource> </InlineEquation> is invertible in <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {M}(\mathcal {B})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">M</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">B</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> if and only if <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$X \notin \mathcal {B}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>∉</mo> <mi mathvariant="script">B</mi> </mrow> </math></EquationSource> </InlineEquation>. Related to the above, we also have multiplier analogs of results of Dyer–Porcelli–Rosenfeld and Aiken, as well as results in the simple C-algebra context.</p>

Perturbations by nilpotent operators in a multiplier algebra

期刊论文
发表日期：

2025年11月25日

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P. W. Ng,
T. Robin,
Arindam Sutradhar

摘要

Let $\mathcal {B}$ $B$ be a separable simple stable purely infinite C*-algebra, and let $\mathcal {M}(\mathcal {B})$ $M (B)$ be the multiplier algebra of $\mathcal {B}$ $B$ . We find a multiplier algebra analog of a result of Brown, Pearcy and Salinas, proving that for all $X \in \mathcal {M}(\mathcal {B})$ $X \in M (B)$ , there exists a nilpotent operator $N \in \mathcal {M}(\mathcal {B})$ $N \in M (B)$ for which $X + N$ $X + N$ is invertible in $\mathcal {M}(\mathcal {B})$ $M (B)$ if and only if $X \notin \mathcal {B}$ $X \notin B$ . Related to the above, we also have multiplier analogs of results of Dyer–Porcelli–Rosenfeld and Aiken, as well as results in the simple C*-algebra context.