# Notions in the literature capturing the “symmetric” or “homogeneous” flavour of $L_p$?

This post/question is admittedly vague, but I hope that with some feedback in comments it could be made more precise.

For $$E$$ a Banach space, $$K(E)$$ and $$B(E)$$ will denote the Banach algebras of compact and bounded linear operators on $$E$$, respectively. There is a tradition / industry in the world of Banach algebras that asks whether $$K(E)$$ or $$B(E)$$ can display various properties if we choose $$E$$ judiciously. As part of this we would like to understand these algebras when $$E$$ is one of the "classical" Banach spaces such as $$\ell_p$$ or $$L_p$$ (with the convention that the unadorned $$L_p$$ is an abbreviation for $$L_p[0,1]$$).

On the other hand, we have made progress in establishing "bad" behaviour of $$B(E)$$ when $$E$$ is a space such as $$\ell_p\oplus \ell_q$$ when $$p\neq q$$, or various $$\ell_p$$-sums of $$\ell_{k(n)}^n$$ ($$n\in {\bf N}$$), not to mention "exotic" examples that have been constructed for the explicit purpose of giving $$B(E)$$ certain properties.

Here are my questions.

Q1. Is there any way to justify, with mathematical definitions or results, the feeling that $$\ell_p$$ and $$L_p$$ are more "symmetrical" examples than the various other examples mentioned above? Note that in various senses $$\ell_p\oplus\ell_q$$ is better behaved than $$L_1$$, e.g. the latter space does not have an unconditional basis, while the former space has a very nice unconditional basis.

Q2. $$\ell_p\oplus\ell_q$$ and the $$\ell_p$$-sums of finite-dimensional $$\ell_k$$ have a flavour of "gluing together things which look different", and hence one might argue that it is not so striking that for such $$E$$ the algebra $$B(E)$$ has behaviour very unlike $$B(\ell_2)$$. Are there established notions in the Banach-space literature that capture the way $$L_p$$ and $$\ell_p$$ look much more "homogeneous"? (The phrase "homogeneous" is meant in a loose sense, not in the sense of https://arxiv.org/abs/math/9205207 )

Regarding Q2: I know that $$\ell_p$$ is a prime Banach space for each $$p$$ (every complemented subspace is isomorphic to the whole space) and that $$L_p$$ is primary (in any direct sum decomposition of the space, at least one of the summands is isomorphic to the whole space), but my understanding is that primeness is too restrictive while being primary allows many more examples. So candidates for answers to Q2 would be notions that are intermediate between "being prime" and "being primary", if such exist.

• A (maybe only loosely related) remark: Within the class of all Banach function spaces, the "symmetric" nature of, for instance, $L^p$ and $\ell^p$ can in a way be described by the notion of a "rearrangement invariant space"; and within the class of all Banach lattices one could perhaps consider a property such as "the group of all isometric lattice isomorphisms acts ideal irreducibly on the given space" to capture the notion of symmetry (although this is admittedly only a vague idea). But I'd guess that you are perhaps more interested in the category of Banach spaces? – Jochen Glueck Jun 9 '19 at 8:27
• Irreducibility of the action of the group of isometries is tempting. However, it has the drawback of being only an isometry invariant, not isomorphism invariant. "Irreducibility of the (linear) isometry group for some equivalent norm" is an isomorphism invariant, but is a less handy notion; I don't know if it's satisfied by $\ell^p\oplus\ell^q$, $p\neq q$. – YCor Jun 9 '19 at 14:10

Fraïssé theory, which identifies structures that are homogenous and universal in some sense, has departed from its model-theoretic roots and is now available in the so-called metric model theory as well as it has a category-theoretic framework developed by W. Kubiś.

In either case, one can show that $$L_p[0,1]$$ is the Fraïssé limit (in a suitable sense) of $$\ell_p^n$$s simply because they can be amalgamated together and stay in the class. In the metric case, this was observed by Ferenczi, Lopez-Abad, Mbombo, and Todorcević (see this recent preprint) and in the category-theoretic setting, this is an unpublished work of Kubiś and myself from 2016 or so. One has to be careful with the exact translation of Fraïssé's theorem as certainly the class of $$\ell_p^n$$-spaces is not closed under taking subspaces so perfect homogeneity is not feasible unless $$p=2$$.

As for your second question, you mentioned primarity. A Banach space $$X$$ is primary whenever $$X = E\oplus F$$, to some closed subspaces $$E,F$$, then at least one of them is isomorphic to $$X$$. Bill Johnson and Detelin Dosev defined the following set:

$$\mathscr{M}_X = \big\{T\in B(X)\colon I_X \neq ATB\; \big(A,B\in B(X)\big)\big\}.$$

This set need not be closed under addition but when it is, it is the unique maximal ideal of $$B(X)$$. For non-primary spaces (that are also isomorphic to their infinite sums, say), the set $$\mathscr{M}_X$$ cannot be closed under addition. The spaces $$\ell_p$$, $$L_p$$, $$C[0,1]$$ are primary, whereas $$\ell_p\oplus\ell_q$$ for $$p\neq q$$ are clearly not.

So perhaps closedness of $$\mathscr{M}_X$$ under addition is the condition you seek?

• Possibly add to your condition that $L(X)/M_X$ is purely infinite. – Bill Johnson Jun 14 '19 at 16:16