I'll use the notion "ordered Banach space" to denote a Banach space $E$ that is ordered by a closed (and convex) cone $E_+$.
Generally speaken, having non-empty interior is not a common property of cones in Banach spaces.
Here are a few details:
Classical function spaces
It is not difficult to see that the usual cone (of elements that are positive almost everywhere) in $L^p(\mathbb{R})$ and $\ell^p$ has empty interior if $p \in [1,\infty)$ and non-empty interior if $p=\infty$.
More generally speaking, if $p \in [1,\infty)$, $(\Omega,\mu)$ is a measure space, and $L^p(\Omega,\mu)$ is infinite-dimensional, then the cone in this space has empty interior. For $p=\infty$ or if the space is finite-dimensional, the cone has non-empty interior (it consists precisely of those function which are larger than an $\varepsilon$ almost everywhere).
For a compact Hausdorff space $K$ the usual cone in the space $C(K)$ of real-valued continuous functions has non-empty interior. It consists precisely of those functions that are $> 0$ everywhere (equivalently, that are larger than an $\varepsilon$ everywhere).
On the other hand, for a locally compact but non-compact Hausdorff space $L$, the standard cone in the space $C_0(L)$ (the space of continuous real-valued functions that vanish at infinity) has empty interior.
Banach lattices
Banach lattices are an abstract generalization of the classical function spaces. Every Banach lattice is an ordered Banach space, but not vice versa. If the cone of a Banach lattice $E$ has non-empty interior, then there exists an equivalent norm which turns $E$ into a so-called AM-space with unit and hence it follows from a representation result of Kakutani that $E$ is isomorphic (as a Banach lattice) to $C(K)$ for a compact Hausdorff space $K$.
This shows that the spaces whose cones have non-empty interior are, in a sense, quite "rare" within all Banach lattices. More specifically, the cone in an infinite-dimensional Banach lattice $E$ has empty interior if, e.g., any of the following conditions is satisfied:
$E$ is (as Banach space) reflexive.
The norm is additive on $E_+$ (i.e., $\|x+y\| = \|x\| + \|y\|$ for all $x,y \in E_+$).
The norm on $E$ is order continuous, i.e. every decreasing net in $E_+$ with infimum $0$ converges to $0$ in norm (this contains the two previous bullet points as special cases).
General ordered Banach spaces
In general ordered Banach spaces the situation becomes a bit more involved.
It is no longer true that, for instance, reflexivity of infinite-dimensional spaces implies that the cone has empty interior.
Here are a few general results about interior points of the cone in ordered Banach spaces. I give a list of examples later on. Let $E$ be an ordered Banach space.
A point $u \in E_+$ is an interior point of $E_+$ if and only if $u$ is a order unit of $E$, which means by definition that for every $x \in E$ there exists a number $\varepsilon > 0$ such that $\varepsilon x \le u$. This and several further characterizations can be found in (warning, self-promotion ahead) [GW20, Proposition 2.11]. However, all those characterizations have been known for decades. We just included them there to have them all in one place (and to have an explicit reference for all those characterizations to which one can point, as I am doing now).
The cone having non-empty interior can be characterized by properties of the duals cone, see for instance [BR84, Section 1.4]. This loosely (well, very loosely) reflects the duality between $L^1$- and $L^\infty$-spaces or between $C^*$-algebras and non-commutative $L^1$-spaces.
If the cone has empty interior (which is, as explained above, quite often the case), there is often still as reasonable replacement for interior points of the cone, namely quasi-interior points and almost interior points. See for instance [Sch74, Section 2.6] and [GW20, Section 2.2].
General ordered Banach spaces: Examples
Here are some concrete examples of ordered Banach spaces:
If $A$ is a $C^*$-algebra and $A_{\operatorname{sa}}$ denotes its self-adjoint part, then $A_{\operatorname{sa}}$ (which is real Banach) is an ordered Banach space when we endow it with the cone of positive semi-definite elements. The cone has non-empty interior if and only if $A$ is unital, and in this case the interior points of the cone are precisely those elements who don't have $0$ in their spectrum. A proof can be found in [GW20, Example 2.15(ii)].
So as a concrete special case, if $H$ is an infinite-dimensional complex Hilbert space, then the cone of positive semi-definite elements within the space $\mathcal{K}(H)_{\operatorname{sa}}$ of all compact self-adjoint operators on $H$ has empty interior.
On the other hand, the cone of positive semi definite elements within the space $\mathcal{L}(H)_{\operatorname{sa}}$ of all bounded self-adjoint operators has non-empty interior and, for instance, the identity operator is an interior point.
In $\ell^2$, consider the so-called ice-cream cone that is given by
$$
K := \{x = (x_n)_{n \ge 0} \in \ell^2: \; x_0 \ge 0 \text{ and } x_0^2 \ge \sum_{n=1}^\infty x_n^2\}.
$$
Then $K$ has non-empty interior (and the interior points are precisely those $x \in K$ for which one has the strict inequality $x_0^2 > \sum_{n=1}^\infty x_n^2$).
On the other hand, if we take the same $K$ as before but now consider it as a subset of $c_0$, then it is still a closed cone, but it is not even generating, i.e., its linear span $K-K$ is not equal to the entire space $c_0$ (since the span is equal to $\ell^2$). In particular, $K$ has empty interior since it is not difficult to show that cones with non-empty interior are always gerating.
The usual cone (pointwise order) in $C^1([0,1])$ has non-empty interior.
Interestingly, if we consider the subspace of functions that vanish at both end points, the cone in this subspace also has non-empty interior. (Indeed, $x \mapsto x(1-x)$ is an interior point of the cone; to see this, use the differentiability of functions $f \in C^1([0,1])$ at the end points of the interval.)
On the other hand, if you consider the subspace $F := \{f \in C^1([0,1]): f(1/2) = 0\}$, then the cone in $F$ has empty interior since it is not even generating (for instance, $x \mapsto x-1/2$ cannot be written as the difference of two positive functions in $F$).
For a bounded open interval $I \subseteq \mathbb{R}$ and $p \in [1,\infty]$ the usual cone (i.e., the cone given by the ordered inherited from $L^p((0,1))$) in the Sobolev space $W^{k,p}((0,1))$ has, for each $k \ge 1$, non-empty interior. However, this is not always true if we replace with $(0,1)$ with a higher dimensional domain.
If we impose $0$ boundary conditions at the end points of the interval, i.e., consider the space $W := \{f \in W^{k,p}((0,1)): f(0) = f(1) = 0\}$, then the cone in $W$ has empty interior if $p < \infty$ and $k=1$ but non-empty interior if $p = \infty$ or $k \ge 2$. The case $k \ge 2$ follows (for all $p \in [1,\infty]$) from the fact that $W$ embeds into $\{f \in C^1([0,1]): f(0) = f(1) = 0\}$ if $k \ge 2$ and that $W$ contains the function $x \mapsto x(1-x)$ from the previous bullet point.
Let $(M,d)$ be a non-empty metric space, $x_0 \in M$ a fixed point, and $\operatorname{Lip}_{x_0}(M)$ the space of all Lipschitz continuous real-valued functions on $M$ that vanish at $x_0$. This is a Banach space when the Lipschitz constant is used as the norm and it is an ordered Banach space when endowed with the pointwise order. The cone has non-empty interior; for instance, the function $x \mapsto d(x,x_0)$ is an interior point.
Literature
For the theory of Banach lattices, there are plenty of books available (although, speaking about order relations, the readability of many of those books is not maximal). When you're interested in interior points of the cone in a Banach lattice, the right notion to look up in those books is "AM-space (or M-space) with unit".
[Sch74] Schaefer: "Banach lattices and positive operators" (1974) [zbMATH]
One of the standard books on the topic, but rather technical and hard to get into. Contains a lot of great results, but good luck finding out there that, say, $L^p$-spaces have order continuous norm if $p < \infty$ if you don't already know it.
Luxemburg and Zaanen: "Riesz spaces I" (1971) [zbMATH]
The standard reference for vector lattices (=: Riesz spaces). Focusses on the algebraic struture without norms.
Zaanen: "Riesz spaces II" (1983) [zbMATH]
Second part of the previously mentioned book. Focusses (in contrast to part I) more on normed vector lattices and Banach lattices.
Meyer-Nieberg: "Banach lattices" (1991) [zbMATH]
In my experience a bit easier to read than the previous ones (though still no fireworks of motivation or context).
Tends to have a somewhat above average number of errors, though most of them are easy to spot when reading with care.
Aliprantis and Burkinshaw: "Positive operators" (1985/2006) [zbMATH]
One of the favourite references for Banach lattices of one of my co-authors. Personally, I haven't used it so much, so I can't really comment on it.
Aliprantis and Border: "Infinite dimensional analysis. A hitchhiker’s guide" (second edition, 1999) [zbMATH]
Contains many results about Banach lattices, embedded into a more general book about functional analysis. Many of the results about Banach lattices are not proved there, but references are given such that one can find the proofs in other books.
In my experience, this is an absolutely great book if one is interested in using Banach lattice theory in other contexts rather than in learning the most subtle technicalities and the most esoteric (counter)examples.
Zaanen: "Introduction to operator theory in Riesz spaces" (1997) [zbMATH]
Delivers what is promised in the title.
Wnuk: "Banach lattices with order continuous norm" (1999) [zbMATH]
An absolutely great resource if you're interested in, well, Banach lattices with order continuous norm. Probably not so relevant for the specific question asked, though, since the cone in a Banach lattice with order continuous norm always has empty interior (unless the space is finite-dimensional).
Regarding general ordered Banach spaces, the literature is much more of a mess. A lot of useful and interesting results, sometimes many decades old, are floating freely through the literature and never made it into a book. Here are the few references that I'm aware of:
Jameson: "Ordered linear spaces" (1971) [zbMATH]
Krasnosel'skii, Lifshits, Sobolev: "Positive linear systems. The method of positive operators" (1989, English translation) [zbMATH]
Unfortunately, the book contains a few rather confusing mistakes. I've been told by somebody who knows the Russian original, though, that the Russian version contains fewer errors. So if you speak Russian it might be worthwhile to read the Russian original instead.
Aliprantis and Tourky: "Cones and duality" (2007) [zbMATH]
Section 2.5 is specifically about ordered Banach spaces.
[BR84] Batty and Robinson: "Positive one-parameter semigroups on ordered Banach spaces" (1984) [zbMATH]
This is actually a lengthy article rather than a book. It consists of two parts; the first part is about ordered Banach spaces, the second part is about positive operator semigroup acting on them.
Vulikh: "Geometrie der Kegel: in normierten Räumen" (2017) [zbMATH]
This is a German translation of two books written in Russian by Vulikh several decades ago. The translation includes some bibliographic updates.
For the older of the two Russian books there also seem to be an English translation from 1967 available [zbMATH].
Kalauch and van Gaans: "Pre-Riesz spaces" (2019) [zbMATH]
This is the most recent book about ordered vector spaces I'm aware of. It does not focus on ordered Banach spaces, but also has some material regarding order units.
Some of the results cited above can be found in the following paper (but as I said, the characterizations of interior points are not due to us; we just gave a summary of what we knew about them, including proofs):
- [GW20] Glück and Weber: "Almost interior points in ordered Banach spaces and the long-term behaviour of strongly positive operator semigroups" (2020) [zbMath]
Some information on interior points and closely related concepts is also discussed in (warning: further self-promotion ahead) a course I gave on ordered Banach spaces in summer 2023. Video recordings of the lectures (in English) are publically available here; see Section 6, which is treated in lecture 20 (mark 54:08) to lecture 22.
