Noncommutative L^p space - Imported Knowlpedia Content

Definition

Let $M$ be a von Neumann algebra, choose a normal semifinite faithful weight $\varphi$ , and form the continuous core $N=M\rtimes_{\sigma^\varphi}\mathbb R$ , a von Neumann crossed product, by its modular automorphism group. Write $\theta$ for the dual action and $\tau$ for the canonical trace on $N$ . For $0<p<\infty$ , the Haagerup noncommutative $L^p$ space is

L^p(M)=\{x:x\text{ is }\tau\text{-measurable and affiliated with }N,\quad \theta_s(x)=e^{-s/p}x\text{ for every }s\in\mathbb R\}.

One sets $L^\infty(M)=M$ . Different choices of $\varphi$ give canonically isometric spaces. The homogeneity condition selects exactly the operators of integrability degree $p$ inside the measurable crossed-product algebra. Here $\tau$ -measurability is computed in the crossed product $N$ .

Independence and basic structure

The crossed product and trace depend on the auxiliary weight, but the homogeneous subspaces do not, up to their canonical identifications. For $1\leq p\leq\infty$ , $L^p(M)$ is a Banach space; $L^1(M)$ identifies isometrically with the predual $M_*$ , and $L^2(M)$ gives the standard-form Hilbert space of $M$ . Multiplication of affiliated operators yields Hölder maps $L^p(M)L^q(M)\subseteq L^r(M)$ when $1/r=1/p+1/q$ Haagerup, pp. 175–184.

Relation to familiar

L^p

spaces

If $M$ is commutative, the construction recovers classical $L^p$ of the corresponding measure class. If $M$ is semifinite with a faithful normal semifinite trace, it is canonically isometric to the tracial noncommutative $L^p$ space. Thus Haagerup's construction extends the tracial theory rather than replacing its formulas in the semifinite case. It remains available for type III algebras, where no faithful normal semifinite trace exists on $M$ itself Terp, Chapters I–II.

Conventions and scope

References

Uffe Haagerup, “ $L^p$ -Spaces Associated with an Arbitrary von Neumann Algebra,” in Algèbres d’opérateurs et leurs applications en physique mathématique, Colloques Internationaux du CNRS 274, CNRS, 1979, 175–184. Institutional scan. Relevant: the crossed-product construction, weight independence, and identifications at $p=1,2,\infty$ .
Marianne Terp, $L^p$ Spaces Associated with von Neumann Algebras, Notes, Mathematical Institute, Copenhagen University, 1981. Institutional scan. Relevant: Chapters I–II on tracial measurable operators and the Haagerup construction.