In this paper, we study algebraic properties of the operations <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$A \circ B=(A \#B)^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>∘</mo> <mi>B</mi> <mo>=</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>#</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$A \diamond B=(A \sigma _{le}B)^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>⋄</mo> <mi>B</mi> <mo>=</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <msub> <mi>σ</mi> <mrow> <mi mathvariant="italic">le</mi> </mrow> </msub> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>, and the squares of Rényi power means on positive definite cones of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$C^*$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-algebras, where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$A \#B$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>#</mo> <mi>B</mi> </mrow> </math></EquationSource> </InlineEquation> denotes the Kubo–Ando geometric mean and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$A \sigma _{le}B$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <msub> <mi>σ</mi> <mrow> <mi mathvariant="italic">le</mi> </mrow> </msub> <mi>B</mi> </mrow> </math></EquationSource> </InlineEquation> denotes the log-euclidean mean of A and B. We are primarily concerned with distributivity and various weaker forms of associativity. We find that these properties are closely related to commutativity and to the centrality of elements in the underlying algebra. We also obtain centrality characterizations concerning the Löwner order.

Algebraic properties of geometric-type operator means and related centrality characterizations

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发表日期：

2025年12月24日

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Richárd Simon

摘要

In this paper, we study algebraic properties of the operations $A \circ B=(A \#B)^2$ $A \circ B = {(A # B)}^{2}$ , $A \diamond B=(A \sigma _{le}B)^2$ $A ⋄ B = {(A σ_{le} B)}^{2}$ , and the squares of Rényi power means on positive definite cones of $C^*$ $C^{*}$ -algebras, where $A \#B$ $A # B$ denotes the Kubo–Ando geometric mean and $A \sigma _{le}B$ $A σ_{le} B$ denotes the log-euclidean mean of A and B. We are primarily concerned with distributivity and various weaker forms of associativity. We find that these properties are closely related to commutativity and to the centrality of elements in the underlying algebra. We also obtain centrality characterizations concerning the Löwner order.