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.