Nonseparable Hilbert spaces

Being nonseparable Banach space is in fact nothing special: one meets the first examples in the standard functional analysis course, when one learns about $\ell^p$ or $L^p[0,1]$ spaces-these spaces are nonseparable when $p=\infty$. However I spoke to one person which said to me that nonseparable Hilbert spaces are rather exotic and in "real life" one rarely come across them. Here I list some examples when I came across nonseparable Hilbert spaces-but to bo honest, none of them had truly convinced me about the importance of those spaces:
1. When $A$ is a $C^*$-algebra, then if $A$ is separable then $A$ embeds into $B(H)$ where $H$ is separable-nevertheless sometimes one is interested in so called universal representation $\pi_u$ of $A$ (one feature of this representation is that any state of $\pi_u(A)$ is in fact a vector state). The Hilbert space of this representation is hardly ever separable.
2. When $Q$ denotes the Calkin algebra (the quotient of $B(H)$ by compact operators) one can show that it cannot be represented on separable Hilbert space (this is very elegant argument: there is an uncountable family of infinite subsets of $\mathbb{N}$ such that the intersection of any two members of this family is finite-this gives an uncountable family of mutually disjoint projections in the quotient -in fact this argument applies also to $\ell^{\infty}/c_0$).
3. There is a notion of almost periodic function on $\mathbb{R}$ (this can be generalized to other groups but let us stick to this particular case) and the space of almost periodic functions contains all functions of the type $e_s(t)=e^{ist}$. One can introduce the scalar product on the linear span of all $e_s$-s and those function turns out to be mutually orthogonal. Completing one obtains a nonseparable Hilbert spaces.
4. There is a notion of tensor product of Hilbert spaces in particular the so called complete tensor product which is due to von Neumann. This construction yields a nonseparable Hilbert space when the tensored family is infinite. But as far as I know, in most application one restricts to the preferred so called $C_0$-sequence thus obtaining the separable space (this is some sort of choosing a ,,stabilization'').

That's all-I don't know any other examples of situations when one is faced with nonseparable Hilbert spaces-so ,,huge'' for the first sight spaces as Fock spaces used in second quantization or $L^2$ over some infinite dimensional spaces with Gaussian measures are all separable. So I would like to ask:

What are some interesting examples of nonseparable Hilbert spaces which you have ever met?

Forgive me if my question is too vaque but I tried to give some motivation around it.

• Related to your example 2: there was some recent work of Ando and Haagerup on ultrapowers arxiv.org/abs/1212.5457 which I think made use of spatial representations of these ultrapowers at various points. Just as with the Calkin, faithful representations of these ultrapowers must live on non-separable Hilbert spaces – Yemon Choi Feb 7 '16 at 0:02
• If you take the universal representation of the weyl algebra instead of the fock representation you get a non-separable Hilbert space – Marcel Bischoff Feb 7 '16 at 2:50

To the contrary, my feeling is that nonseparable Hilbert spaces are in some sense artifacts and can almost always be avoided. And more generally, to your comment about nonseparable Banach spaces being "nothing special", most of the nonseparable Banach spaces one meets in practice are duals of separable Banach spaces, and therefore are weak* separable. Algebras of almost periodic functions, and their noncommutative analogs, the CCR algebras, are rare examples of interesting nonseparable Banach spaces which are not dual spaces, but even here one is mainly interested in regular representations, which are determined by their behavior on separable subalgebras (the span of the functions $e_s$ with $s$ rational, say).

@Yemon Choi: the answer to your comment is too long for a reply so putting it in another answer to the original question – hope this is not off-topic for the original post.

The space $${\mathbb{R}}^{[0,1]}$$ and the $$L^2$$ functions on it arise not as a model of how the brain actually works, but as a counterfactual, pathological example.

In neuroscience, a sensory stimulus (for example an image seen by the eye) causes a pattern of activity in a set of neurons, known as a neural code for the stimulus. We can model this activity using a real-valued function $$f(n,s)$$, where $$n$$ represents the neuron, and $$s$$ represents the stimulus. (The same holds for artificial neural networks used in machine learning.) The actual brain of course contains a finite number of neurons, but to make mathematical models it is useful to consider them as drawn randomly from a continuous probability distribution, and the same for the stimuli.

A classical hypothesis of neuroscience, originally inspired by information theory, holds that the brain would operate most efficiently if neural responses were independent. However if you require the responses of all neurons be independent whatever the stimulus, you soon end up considering examples like neurons as random functions drawn from $${\mathbb{R}}^{[0,1]}$$ with the product Gaussian measure, and $$f(n,s)$$ as evaluation at the point $$s\in[0,1]$$. This is a pathological model, as such functions are almost surely discontinuous, and it is also not consistent with experimental brain recordings. You are correct that stochastic processes are a much better model for how the brain actually works, and brain recordings actually let us further constrain the smoothness of these stochastic processes. Nevertheless it is still useful to consider counterfactual models such as the above. The paper I cited deals with the smoothness constraints, and does not yet mention the counterfactual, but a forthcoming revision will.

• why not use white noise instead ?encyclopediaofmath.org/index.php/White_noise_analysis – Abdelmalek Abdesselam Jan 3 at 17:21
• Thanks for the suggestion! I'm not familiar with white noise analysis, but if I understand correctly random samples of white noise will almost surely not be real-valued functions, but rather distributions which can't be evaluated at a point. In which case it's not clear how to apply them as a model. – Neuromath Jan 3 at 18:07

Let A = $$\mathbb{R}^{[0,1]}$$, with a product Gaussian measure - in other words, consider an independent Gaussian-distributed random variable for each real number in $$[0,1]$$. The space $$L^2[A]$$ is non separable.

Believe it or not, this example has come up in neuroscience.

• I'd rather not have to believe it... can you provide a reference? – Wojowu Dec 31 '18 at 11:49
• Well, it's actually in my own research, not yet published - you can find a preliminary version that does not mention separability or this particular example here - for the math, skip straight to appendix 2 at the end. For a consideration of non-separable Hilbert spaces in machine learning, see here. – Neuromath Dec 31 '18 at 16:03
• Where in Appendix 2? One would normally expect to be taking some kind of Gaussian process indexed by ${\bf R}$ equipped with its usual metric, so (admittedly speaking as a non-probabilist) I find it surprising that one would need the space A that you mention – Yemon Choi Dec 31 '18 at 22:38