Nonseparable counterexamples in analysis

Question

When asking for uncountable counterexamples in algebra I noted that in functional analysis there are many examples of things that “go wrong” in the nonseparable setting. But most of the examples I'm thinking of come from $\mathrm C^*$-algebras (equivalent formulations of being type I; anything having to do with direct integral decompositions; primeness versus primitivity, etc.).

Question. Are there also good examples from Banach space theory, or from harmonic analysis? Or from analysis in general? (Or even other examples from $\mathrm C^*$-algebra that I might not know about?)

Answer 1

Sobczyk's theorem: if $Z$ is a subspace of a separable Banach space $X$ that is isomorphic to $c_0$, then $Z$ is complemented in $X$, fails for many non-separable spaces such as $X=\ell_\infty\cong C( \beta \mathbb N)$.

For related reasons, the Borsuk–Dugundji extension theorem (which says that if $F$ is a closed, metrisable subspace of a compact space $K$, then you can apply the Tietze–Urysohn extension theorem in a linear way, i.e., there is a contractive operator $T\colon C(F)\to C(K)$ such that $(Tf)|_F=f$ for $f\in C(F)$) fails when $F$ is non-metrisable, that is, $C(F)$ is non-separable in the uniform norm. (Here, $F = \beta \mathbb N\setminus \mathbb N \subset \beta\mathbb N$ is the easiest counterexample.)

Answer 2

Probability theory can get pretty weird in the non-separable case. For example, write $B$ for the non-separable Banach space of sequences $a = (a_k)_{k \ge 1}$ such that $$ |a|^2 := \sup_{n \ge 0} 2^{-n} \sum_{k=2^n}^{2^{n+1}} |a_k|^2 < \infty\;. $$ Take now an i.i.d. sequence of Gaussian variables $\xi = (\xi_k)_{k \ge 1}$. Then, one has $|\xi| < \infty$ almost surely but, for any fixed $a \in B$, one also has $P(|\xi-a| \ge 1) = 1$, so the notion of "support" becomes pretty problematic...

Answer 3

Farah and Ozawa showed that there are uncountable (free) abelian groups $\Gamma$ which admit bounded homomorphisms $\theta:\Gamma \to {\rm Inv}{\mathcal Q}(\ell^2)$, ${\mathcal Q}$ being the Calkin algebra, which are not unitarizable: that is, there is no invertible element of the Calkin which conjugates $\theta$ to a unitary representation. (See the original preprint by Farah and Ozawa: arXiv 1309.2145v1.)

One can refine their original construction so that $\Gamma$ is a direct sum of uncountably many copies of ${\mathbb Z}/2{\mathbb Z}$ and the homomorphism $\theta$ takes values in the $(\ell^\infty/c_0)\otimes {\bf M}_2$ (see the "later editions" of the aforementioned preprint, via arXiv 1309.2145).

In both cases, if one restricts attention to countable abelian (or even amenable) groups then unitarization is possible by a variation on the classical averging argument of Dixmier, Day and others.

Answer 4

The norm of (even a continuous convex function on) a separable Banach space is Gâteaux differentiable on a dense $G_\delta$-set (Mazur), but the canonical norm on the nonseparable Banach space $\ell_1(\Gamma)$, $\Gamma$ uncountable, is nowhere Gâteaux differentiable.

Answer 5

One situation which highlights the difference between separability and non separability is that of an important class of locally convex spaces which fail many of the properties which are relevant to closed graph theorems (barrelled, bornological, etc.) but which do satisfy such a result for mappings into SEPARABLE spaces. This has interesting consequences, one of which we will now sketch.

A fascinating phenomenon involving non separability is the fact that many classical Banach spaces consist of bounded objects (continuous, uniformly continuous, measurable functions, linear operators) with the supremum norm and such spaces are ALWAYS non separable, or rather only separable under tight conditions. Thus separability of $C(K)$ implies that $K$ is metrisable, separability of $\ell^\infty$ requires finite dimensionality as does that of $L(H)$ and so on. There is a meta explanation of this fact which goes back to work of Saks, more precisely his original version of what is now known as the Vitali-Hahn-Saks theorem. Saks used a Banach-Steinaus-type theorem (rather than a closed graph theorem but these two themes are known to be closely related). The novelty of his approach lay in his use of the Baire category theorem (what else?), not on a Banach space but on the unit ball of an $L^\infty$ space topologised by the $L^1$ norm. The fact that the latter is not a linear space and the importance of translation in the Banach case made the proof more delicate and it required a geometrical condition relating the norms, now called $\Sigma$.

This led to the theory of Saks spaces—a unified approach to a class of Banach function or operator spaces for which the fact that the norm was too strong for certain applications (dual too large, bad density properties...) was remedied by replacing it with a weaker, but in these respects more suitable, l.c. topology (two-normed spaces, mixed topologies, Saks spaces ...). The paradigmatic example is, perhaps, the strict topology on the space of bounded, continuous functions on a locally compact space (R. C. Buck, et al.)—the elements of its dual space are precisely the bounded, tight measures.

The relevance of this to the question on the table is that one can use Saks’ ideas to obtain a closed graph theorem for linear mappings from certain Saks spaces into SEPARABLE Banach spaces. This means that in the concrete examples of interest, separability of the original Banach space forces a kind of collapse—the Saks space structure coincides with the Banach space one and this can only happen under restricted conditions.

Details are in the book ”Saks Spaces and Applications to Functional Analysis” which can be found online.

Answer 6

When $1< p< \infty$, the space $L_p(0,1)$ has an unconditional basis (the Haar basis). Enflo and Rosenthal proved that for $p\not=2$ the space $L_p(0,1)^\gamma$ does not have an unconditional basis when $\gamma \ge \omega_\omega$.

Answer 7

As seen in some contributions here, non separability of a Banach space can lead to pathology in the properties of measures or measurable functions with values therein. As an example of this phenomenon, I would mention Stegall’s result that a non-separable dual of a separable Banach space fails the Radon Nikodym property. The construction that he used even charactises dual spaces $E’$ with RNP as those for which separable subspaces of $E$ always have separable duals. The details are in the classic text by Diestel and Uhl.

Answer 8

Every separable Banach space is bi-Lipschitz isomorphic to a subset of $c_0$. But there are Banach spaces that do not uniformly embed in $c_0(\Gamma)$ for any $\Gamma$. See

• Jan Pelant, Petr Holický, Ondřej F. K. Kalenda, $C(K)$ spaces which cannot be uniformly embedded into $c_0(\Gamma)$, Fundamenta Mathematicae 192 (2006) pp. 245–254 (journal abstract page).

Answer 9

For AF C*-algebras, there are a number of differences between separable and nonseparable algebras, and something weird happens at $\aleph_2$, too.

Recall that an AF (C*)-algebra is the norm-closure of a directed limit of finite-dimensional C*-algebras (the norm is uniquely determined), equivalently, for every $\epsilon > 0$, any finite set of elements can be approximated to within $\epsilon$ by elements of a finite-dimensional algebra.

George Elliott proved that for separable AF algebras (the usual kind), ordered pointed K$_0$ is a complete invariant (for isomorphism), and Effros Handelman and Shen determined exactly which partially ordered abelian groups can so arise; these are known as dimension groups (partially ordered abelian gps satisfying Riesz interpolation and unperforation)

When the locally semisimple subalgebra has first uncountable dimension (which corresponds to the smallest nonseparability condition for the C*-algebra), it was already known (from the 40s, in a Russian paper) that ordered pointed K$_0$ is not complete. However, it was shown (by Goodearl and Handelman) that every first uncountable dimension group could arise as the ordered K$_0$ of an AF algebra with first uncountable dimensional locally semisimple algebra.

However, when the dimension group is of cardinality $\aleph_2$, it was shown (by a well-known mathematician whose name escapes me at the moment), that there exist dimension groups of this cardinality which cannot be realized by a suitable direct limit, hence cannot be realized by corresponding AF algebras.