General theory for p-normed spaces

Question

Hello,

in functional analysis and operator theory, you encounter several (at first glance, at least) similar constructions of normed spaces that can be indexed with some $ p \in [1,\infty]$, and which behave very similarly. These include

• $\mathbb R^n$ with a $p$-norm
• $l^p$ spaces
• $L^p$ spaces
• $p$-Trace-norms on matrices
• $p$-Schatten-classes

Are you aware of an approach that handles these objects in an unified abstract way? An interesting result would be the construction of such a family of spaces from a given Hilbert space.

Answer 1

As Nate and I pointed out in comments, your question reduces to asking whether there is a unified framework which includes both $L^p$ spaces and Schatten spaces. One such framework (there may be others) is noncommutative $L^p$ spaces. (Usual $L^p$ spaces are the commutative special case.) There's a nice survey article by Pisier and Xu in the Handbook of the Geometry of Banach Spaces.

Answer 2

To supplement Mark Meckes' answer let me point out that all of your five examples are L^p-spaces of the corresponding von Neumann algebras:

• bounded functions on the disjoint union of n points for R^n;
• bounded functions on the countable disjoint union of points for l^p;
• bounded functions on the underlying measurable space of a smooth manifold for L^p;
• type I_n factors for matrices;
• type I_∞ factor for Schatten classes.

All these algebras are type I von Neumann algebras. L^p spaces for type II and type III algebras are much more interesting (think of Tomita-Takesaki theory etc.).

Moreover, there is no need to confine 1/p to the interval [0,1]. In fact, 1/p can be be any complex number with a nonnegative real part. In particular, L^p spaces for imaginary 1/p are used in Tomita-Takesaki theory.

Furthermore, the theory can be extended to modules over von Neumann algebras. In particular, we can apply this theory to the case of Hilbert spaces (i.e., modules over the von Neumann algebra of complex numbers). However, we do not obtain anything new in this case because L^p spaces of the von Neumann algebra of complex numbers are canonically isomorphic to the vector space of complex numbers.

For the reference I recommend “Algebraic aspects in modular theory” by Shigeru Yamagami. In my opinion it's easier to read because its approach is mostly algebraic. Other papers are usually very technical and lean towards analysis, not algebra.

Answer 3

This is probably not quite an answer to your question but rather a hint how a generalization should look like. As already mentioned in the comments, the discrete and continuous measure-theoretic $\ell^p$ spaces are just one and the same thing, depending only on the kind of measure space you consider. Also the $p$-Schatten classes generalize those for finite-dimensional matrices. In fact, Schatten classes can be defined by means of approximation numbers not refering to the trace. This approach makes this notion available to continuous endomorphisms of Banach spaces as well. A nice reference for this is e.g. Jarchow's book on locally convex analysis...

In noncommutative geometry one tries to unify also the measure-theoretic and the operator algebraic versions of $\ell^p$: one considers the later ones as $L^p$ "functions" on a "noncommutative measure space". There are von Neumann algebraic versions of the Schatten classes around which generalize the Schatten classes even further. I guess that this is the unified framework you are looking for.I'm not really familiar with the literatur in detail, but there are also noncommutative versions of measure theoretic theorems like the Radon-Nikodym theorem etc. A look at Connnes Noncommutative Geometry will certainly help...