We present a generalization of Hölder duality to algebra-valued pairings via <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-modules. Hölder duality states that if <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$p \in (1, \infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$p'$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>p</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation> are conjugate exponents, then the dual space of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L^p(\mu )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is isometrically isomorphic to <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L^{p'}(\mu )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <msup> <mi>p</mi> <mo>′</mo> </msup> </msup> <mrow> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. In this work, we study certain pairs <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$(\textsf{Y},\textsf{X})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="sans-serif">Y</mi> <mo>,</mo> <mi mathvariant="sans-serif">X</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, as generalizations of the pair <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$(L^{p'}(\mu ), L^p(\mu ))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msup> <mi>L</mi> <msup> <mi>p</mi> <mo>′</mo> </msup> </msup> <mrow> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, that have an <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-operator algebra-valued pairing <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\textsf{Y}\times \textsf{X}\rightarrow A$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="sans-serif">Y</mi> <mo>×</mo> <mi mathvariant="sans-serif">X</mi> <mo stretchy="false">→</mo> <mi>A</mi> </mrow> </math></EquationSource> </InlineEquation>. When the A-valued version of Hölder duality still holds, we say that <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$(\textsf{Y}, \textsf{X})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="sans-serif">Y</mi> <mo>,</mo> <mi mathvariant="sans-serif">X</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is C-like. We show that finite and countable direct sums of the C-like module (A, A) are still C*-like when A is any block diagonal subalgebra of <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$d \times d$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>×</mo> <mi>d</mi> </mrow> </math></EquationSource> </InlineEquation> matrices. We provide counterexamples when <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$A \subset M_d^p(\mathbb {C})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>⊂</mo> <msubsup> <mi>M</mi> <mi>d</mi> <mi>p</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is not block diagonal.

C*-like modules and matrix p-operator norms

期刊论文
出版模式：

开放获取
发表日期：

2026年3月13日

查看全文

Alessandra Calin,
Ian Cartwright,
Luke Coffman,
Alonso Delfín,
Charles Girard,
Jack Goldrick,
Anoushka Nerella,
Wilson Wu

摘要

We present a generalization of Hölder duality to algebra-valued pairings via $L^p$ $L^{p}$ -modules. Hölder duality states that if $p \in (1, \infty )$ $p \in (1, \infty)$ and $p'$ $p^{'}$ are conjugate exponents, then the dual space of $L^p(\mu )$ $L^{p} (μ)$ is isometrically isomorphic to $L^{p'}(\mu )$ $L^{p^{'}} (μ)$ . In this work, we study certain pairs $(\textsf{Y},\textsf{X})$ $(Y, X)$ , as generalizations of the pair $(L^{p'}(\mu ), L^p(\mu ))$ $(L^{p^{'}} (μ), L^{p} (μ))$ , that have an $L^p$ $L^{p}$ -operator algebra-valued pairing $\textsf{Y}\times \textsf{X}\rightarrow A$ $Y \times X \to A$ . When the A-valued version of Hölder duality still holds, we say that $(\textsf{Y}, \textsf{X})$ $(Y, X)$ is C*-like. We show that finite and countable direct sums of the C*-like module (A, A) are still C*-like when A is any block diagonal subalgebra of $d \times d$ $d \times d$ matrices. We provide counterexamples when $A \subset M_d^p(\mathbb {C})$ $A \subset M_{d}^{p} (C)$ is not block diagonal.