math.OA — Operator Algebras

Universal Modules, Natural Pairings, and Weighted Isometries for Noncommutative $L^p$-Spaces

Contributed by Zhenyu Sun, Jilin Wang

For $1\le p<\infty$, we consider the functor $L^p$ on finite tracial von Neumann algebras with trace-preserving embeddings, including nonunital ones. For a functor of unital contractive left modules with compatible distinguished vectors $\xi_M$, finite direct sums of scalars determine the best constant in $\|x\xi_M\|\le K\|x\|_p$. The pointed $L^p$ functor is initial among the families satisfying this estimate with $K=1$. Retaining only the bound on scalar points gives an atomic norm. On $\mathbb C^n$ with unit weights the norm of the identity from $\ell^p_n$ to this atomic space is asymptotic to $p^{-1}(\log n)^{1-1/p}$ for $p>1$. For $1\le p,q<\infty$, every objectwise bounded natural bilinear family on $L^p\times L^q$ has the form $c\,\tau(xy)+d\,\tau(x)\tau(y)$ for $x,y\in M$, with $c,d$ independent of the object and $c=0$ when $1/p+1/q>1$. A uniformly bounded natural inner product on every $L^p(M,\tau)$ exists exactly at $p=2$. For finite-dimensional unital embeddings preserving a faithful positive functional, we determine the nearest modularly compatible target density both in Hilbert--Schmidt norm and in the Hilbert--Schmidt distance between positive square roots. The minimum squared distance between square roots lies between a weighted sum of the losses in squared $L^2$ norm on matrix units and twice that sum. Both constants are sharp.