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Motivation and background This question is motivated by the problem of classifying the (two-sided) closed ideals of the Banach algebra $\mathcal{B}(L_\infty)$ of all (bounded, linear) operators on $L_\infty$ $(=L_\infty[0,1])$. As far as I am aware, the only known nontrivial ideals in $\mathcal{B}(L_\infty)$ are the ideals $\mathcal{K}(L_\infty)$ of compact operators and $\mathcal{W}(L_\infty)$ of weakly compact operators. Most of the other well-known closed operator ideals not containing the identity operator of $L_\infty$ seem to coincide with one of these two operator ideals on $L_\infty$. Let me mention explicitly the following further relevant pieces of background information:

  • Any nontrivial closed ideal $\mathcal{J}$ of $\mathcal{B}(L_\infty)$ must satisfy $\mathcal{K}(L_\infty)\subseteq \mathcal{J} \subseteq \mathcal{W}(L_\infty)$.

  • We have $\mathcal{S}(L_\infty) = \mathcal{W}(L_\infty)$, where $\mathcal{S}$ denotes the (closed) operator ideal of strictly singular operators. Moreover, the ideal of operators $L_\infty \longrightarrow L_\infty$ that factor through $L_2$ $(=L_2[0,1])$ is a norm dense subset of $\mathcal{W}(L_\infty)$. With regards to this latter fact, we have that if $\mathcal{J}$ is any closed ideal of $\mathcal{B}(L_\infty)$ satisfying $\mathcal{K}(L_\infty)\subsetneq \mathcal{J} \subsetneq \mathcal{W}(L_\infty)$, then there exists an element of $\mathcal{W}(L_\infty)\setminus \mathcal{J}$ that factors through $L_2$.

One possible avenue towards discovering more closed ideals in $\mathcal{B}(L_\infty)$ is to consider products of closed operator ideals, and my question below concerns but one approach along these lines. We recall now that for operator ideals $\mathcal{A}$ and $\mathcal{B}$, their product $\mathcal{B}\circ\mathcal{A}$ is the class of all operators of the form $BA$, where $A\in\mathcal{A}$, $B\in\mathcal{B}$ and the codomain of $A$ coincides with the domain of $B$ (so that the composition is defined). It is well-known that $\mathcal{B}\circ\mathcal{A}$ is an operator ideal and that $\mathcal{B}\circ\mathcal{A}$ is a closed operator ideal whenever $\mathcal{A}$ and $\mathcal{B}$ are, the latter fact being due to Stefan Heinrich. For $n$ a natural number, one may define the power of an operator ideal $\mathcal{A}^n$ as the product $\mathcal{A}\circ \ldots \circ \mathcal{A}$ with $n$ factors, and $\mathcal{A}^n$ is closed for all $n$ whenever $\mathcal{A}$ is closed. Moreover, we may define $\mathcal{A}^\infty:= \bigcap_{n\in\mathbb{N}}\mathcal{A}^n$, with $\mathcal{A}^\infty$ being a closed operator ideal whenever $\mathcal{A}$ is.

It is well-known that every operator from $L_\infty$ to $L_2$ $(=L_2[0,1])$ is strictly singular. However, the answer to the following question is not so clear to me:

Question: Is $\mathcal{B} (L_\infty, L_2) = \mathcal{S}^\infty (L_\infty, L_2)$? If no, what is the least $n$ for which $\mathcal{B} (L_\infty, L_2) \neq \mathcal{S}^n (L_\infty, L_2)$?

Further comments: A negative answer to my question above will yield through easy arguments that $\mathcal{K}(L_\infty) \subsetneq \mathcal{S}^\infty (L_\infty) \subsetneq \mathcal{W}(L_\infty)$ (note that $\mathcal{K}(L_\infty) \subsetneq \mathcal{S}^\infty (L_\infty)$ in any case since there are noncompact operators $L_\infty \longrightarrow L_\infty$ that have the formal inclusion operator $\ell_2 \hookrightarrow \ell_\infty$ as a factor, and this inclusion operator belongs to $\mathcal{S}^\infty$).

It seems to me that the best chance of obtaining an element of $\mathcal{B} (L_\infty, L_2) \setminus \mathcal{S}^\infty (L_\infty, L_2)$ would be to consider a surjective element of $\mathcal{B} (L_\infty, L_2)$, for instance the adjoint of an isomorphic embedding of $L_2$ into $L_1$. A further possibility along these lines would be to factor such an embedding as the product of $n$ operators whose adjoints are strictly singular, for each $n\in \mathbb{N}$. However, it seems to me that such an approach will require a far deeper knowledge of the subspace structure of $L_1$ than I have at the present time. Interpolation methods may work too, but I am also too ignorant of that theory to know whether it's a genuinely feasible approach.

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Nice question, Phil. So it boils down to whether $I_{\infty,2}$ is the product of strictly singular operators, where $I_{\infty,2}$ is the identity from $L_\infty(\mu)$ to $L_2(\mu)$ with $\mu$ a probability (since every operator from $L_\infty$ to $L_2$ is $2$-summing). –  Bill Johnson Jun 16 '10 at 8:49
    
Thanks Bill, that's a very good point. I wonder then whether or not $I_{\infty, 2}$ is the product of two $\ell_2$-singular operators? –  Philip Brooker Jun 16 '10 at 12:52

1 Answer 1

up vote 5 down vote accepted

I think this is an outline of a proof, Phil. It is enough to factor $I_{\infty,2}$ as the product of strictly singular operators, where $I_{\infty,2}$ is the identity from $L_\infty(\mu)$ to $L_2(\mu)$ with $\mu$ a probability (since every operator from $L_\infty$ to $L_2$ is $2$-summing). I guess it can be assumed that $\mu$ is a separable measure (if not, I think using Maharam's theorem works) and certainly it is enough to look at $\mu$ purely non atomic. But then $I_{\infty,2}$ can be regarded as the identity from $L_\infty([0,1])$ to $L_2([0,1])$ since every separable non atomic probability space is measure isometric to $[0,1]$ with Lebesgue measure. Now $I_{\infty,2}$ factors through $L_\Phi([0,1])$ for every fast growing Orlicz function $\Phi$. Consider $\Phi_p(t)=e^{t^p} -1$ for $p>$. Rodin and Semenov showed that the identity from $L_{\Phi_p}$ to $L_2$ is strictly singular and the injection from $L_\infty$ to $L_\Phi$ is strictly singular always. This takes care of the case $n=2$.

Schechtman suggests that to do general $n$ to show that the injection from $L_{\Phi_p}$ to $L_{\Phi_r}$ is strictly singular for $2<r<p$. At least this map is not an isomorphism on the span of the Rademachers.

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Thanks Bill, that looks like it should work, and in fact suggests a general method for ruling out other products of operator ideals as giving new closed ideals in $\mathcal{B}(L_\infty)$, e.g. finitely strictly singular operators. Plus, as an added bonus, it gives me a good incentive to go and learn more about Orlicz spaces. –  Philip Brooker Jun 18 '10 at 1:17
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@Philip Booker This article takes care of the case of general $n$ as per Schechtman's suggestion: MR2515584 (2010d:46025) 46B70 (46E30) Astashkin, Sergei V. (RS-SAMA) ; Herandez, Francisco L. (E-MADC) ; Semenov, Evgeni M. [Semenov, Evgenı Mikhılovich] (RS-VORO) –  Bill Johnson Jun 21 '10 at 8:32
    
Thanks again Bill, that answers my question very nicely. I really appreciate your answer and comments. –  Philip Brooker Jun 23 '10 at 1:23

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