Tracial noncommutative L^p space - Imported Knowlpedia Content

Definition

Let $M$ have a faithful normal semifinite trace $\tau$ , and let $0<p<\infty$ . The tracial noncommutative $L^p$ space is

L^p(M,\tau)= \left\{x\in S(M,\tau):\tau(|x|^p)<\infty\right\},

where $S(M,\tau)$ is the algebra of $\tau$ -measurable operators affiliated with $M$ . Its $p$ -size is $\lVert x\rVert_p=\tau(|x|^p)^{1/p}$ . For $p\geq1$ this is a complete norm; for $0<p<1$ it is a complete quasinorm. By convention $L^\infty(M,\tau)=M$ with the operator norm.

Products, duality, and interpolation

Noncommutative Hölder inequality gives

\lVert xy\rVert_r\leq\lVert x\rVert_p\lVert y\rVert_q \quad\text{when}\quad \frac1r=\frac1p+\frac1q.

For $1\leq p<\infty$ and conjugate exponent $q$ , the pairing $(x,y)\mapsto\tau(xy)$ identifies the Banach dual of $L^p(M,\tau)$ with $L^q(M,\tau)$ , with the usual endpoint interpretation. The spaces also form the expected complex interpolation scale. These results extend classical $L^p$ inequalities to noncommuting products Fack–Kosaki, §4.

Canonical examples

For $M=L^\infty(X,\mu)$ and $\tau(f)=\int_X f\,d\mu$ , the construction is the classical $L^p(X,\mu)$ . For $M=B(H)$ with its usual trace, it is the Schatten class: compact operators whose singular values form an $\ell^p$ -sequence. In a finite von Neumann algebra with normalized trace, every bounded element belongs to every finite $L^p$ , and completing the bounded finite-trace ideal in $\lVert\cdot\rVert_p$ gives the same space Nelson, pp. 107–116.

Dependence and scope

References

Edward Nelson, “Notes on Non-Commutative Integration,” Journal of Functional Analysis 15 (1974), 103–116. DOI record. Relevant: pp. 107–116 on integration and $L^p$ -type spaces for a semifinite trace.
Thierry Fack and Hideki Kosaki, “Generalized s-Numbers of $\tau$ -Measurable Operators,” Pacific Journal of Mathematics 123 (1986), 269–300. DOI record. Relevant: §§3–4 on measure convergence, $L^p$ norms, and Hölder-type inequalities.