It is well known that both the sequence spaces $c$ and $c_0$ have duals which are isometrically isomorphic to $\ell^1$. Now, $c_0$ is a subspace of $c$. My question – is there an even smaller subspace of $c$ whose dual is isometrically isomorphic to $\ell^1$? More generally, is there a characterization of preduals of $\ell^1$? References are most welcome.
Functional Analysis – Spaces Smaller Than $c_0$ with Dual $\ell^1$
banach-spacesfunctional-analysis
Best Answer
Preduals of $\ell_1$ are very interesting creatures. The question is slightly ill-posed but let me comment on that anyway.
We have quite a lot of isometric preduals of $\ell_1$. Indeed, for any countably infinite ordinal number $\alpha$ (endowed with the order topology) we have $C_0(\alpha)^* \cong \ell_1$. This essentially follows from the Riesz–Markov–Kakutani representation theorem as every measure on such a space has to be atomic.
Note that $c_0 = C_0(\omega)$ and $c=C(\omega+1)$ are isomorphic. Actually for each $\alpha\in [\omega, \omega^\omega)$ we have $C_0(\alpha) \cong c_0$. However, there exist $\aleph_1$ many countable ordinals which give pair-wise non-isomorphic Banach spaces. To be more precise, if $\alpha, \beta$ are countable ordinals then $C(\omega^{\omega^\alpha}+1)$ and $C(\omega^{\omega^\beta}+1)$ are isomorphic if and only if $\alpha = \beta$.
Historically, the first example different from the above-mentioned ones was due to Y. Benyamini and J. Lindenstrauss:
Johnson and Zippin proved that every isometric predual is a quotient of $C(\Delta)$, where $\Delta$ is the Cantor set.
This result combined with an old result of Pełczyński
asserting that an operator $T\colon C(K)\to X$ is weakly compact if and only if it is not bounded below on any isomorphic copy of $c_0$ yields the following corollary:
Corollary. Each isometric predual of $\ell_1$ contains a subspace isomorphic to $c_0$.
In some sense this answers OP's question.
One may wonder whether the Cantor set in the statement of the Johnson–Zippin theorem may be replaced by a countable compact Hausdorff space. This is not the case as shown by D. Alspach:
On the other hand, Gasparis constructed some new preduals of $\ell_1$ which are quotients of $C(\alpha)$:
Since there is no characterisation of complemented subspaces of $C(K)$-spaces, this result is also noteworthy:
If you are interested in spaces whose dual space is only isomorphic to $\ell_1$ then the situation is even more exciting. J. Bourgain and F. Delbaen constructed isomorphic preduals without subspaces isomorphic to $c_0$:
A recent variation of the BD construction is the famous Argyros–Haydon space which is an isomorphic $\ell_1$-predual with the property that each operator on this space is of the form $cI + K$, where $c$ is a scalar and $K$ is a compact operator.
Even more recently, Spiros A. Argyros, Ioannis Gasparis and Pavlos Motakis constructed a $c_0$-asymptotic $\ell_1$-predual without copies of $c_0$:
A nice overview of the BD-spaces (and some new variants of the Argyros–Haydon space) can be found in Matt Tarbard's PhD thesis:
I would also recommend the following paper by M. Daws, R. Haydon, Th. Schlumprecht and S. White
which explains these subtleties very well.
Laustsen and myself have exhibited a version of the Argyros–Haydon space whose algebra of bounded operators has certain peculiar properties:
Let me finish with an open problem (I think) which I like very much:
Problem. We can easily construct $\aleph_1$ many isometric preduals of $\ell_1$. Can we construct in $\mathsf{ZFC}$ continuum many isometric preduals of $\ell_1$?