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17h
answered Compact non-nuclear operators
1d
comment Two questions on the James $p$-space $J_{p}(1<p<\infty)$
Q 2: It is elementary that a weakly sequentially complete space that has separable dual is reflexive.
1d
comment Two questions on the James $p$-space $J_{p}(1<p<\infty)$
Q 1: Doesn't Andrew's proof show that $z_n^*$ is equivalent to the unit vector basis of $\ell_q$, where $q$ is the conjugate index to $p$?
Apr
20
comment $l^1$ versus $l^2$
Both $L_1$ and $\ell_1$ are paved by $\ell_1^n$-s, Nik. I think Denis was making the point that Dvoretzky's Theorem for $L_1$ spaces goes way back. Another way to see that is to observe that $L_1^{2^n}$ contains a length $n$ sequence of IID Bernoulli random variables and use Berry-Essen (or even just CLT in the right form) to get random variables that approximate IID N(0,1) random variables. It is basically obvious that IID N(0,1) random variables are, in the $L_1$ norm, isometrically equivalent to an orthogonal sequence in a Hilbert space.
Apr
18
comment $l^1$ versus $l^2$
If $\mu$ is not purely atomic, then $L_1(\mu)$ contains a subspace isometrically isomorphic to $\ell_2$. Nik's question was for the case where $\mu$ is purely atomic.
Apr
18
comment $l^1$ versus $l^2$
It is not a fact, Nik: Given any $\varepsilon>0$ and any infinite dimensional Banach space $X$, there is a norm $|\cdot |$ on $X$ that is equivalent to the original norm and, for all $n$, $\ell_2^n$ embeds isometrically into $(X,|\cdot |)$. This is a consequence of Dvoretzky's Theorem. I think you can even make the $|\cdot |$ norm $1+\epsilon$-equivalent to the original norm, thought that would take a bit more effort.
Apr
13
comment Unconditionally $p$-converging operators on $L_{1}[0,1]$
I don't know the answer.
Apr
12
answered Unconditionally $p$-converging operators on $L_{1}[0,1]$
Apr
4
comment is every element in a C* algebra a product of normal elements?
Interesting addition. I don't see that it suggests a classification of the operators on $\ell_2$ that are products of normal operators.
Apr
4
comment is every element in a C* algebra a product of normal elements?
Is there a classification of operators on $\ell_2$ that are products of normal operators?
Mar
25
comment Maurey-Pisier Theorem for complex Banach spaces
Did you try to fill in the details in my answer to that question? That gives only $4+\epsilon$ finite representability, but I suggested how you might get $1+\epsilon$ finite representability.
Mar
24
answered Density of sets whose image is dense
Mar
19
comment A Banach space with the BD property and without the weak Gelfand-Phillips property
You should correct your definition of wGP.
Mar
19
comment Non-reflexive Orlicz spaces
$805.92?!?!?!?!
Mar
16
comment Classification of subsymmetric basic sequences
Yes, it is trivial that a subsymmetric basic sequence is bounded and bounded away from zero. If a basis is subsymmetric and not weakly null, then there is a bounded linear functional that is bounded away from zero on a subsequence of the basis, hence, by subsymmetry, there is another bounded linear functional that is bounded away from zero on the basis itself. Unconditionality then gives that the basis is equivalent to the unit vector basis of $\ell_1$.
Mar
16
answered Classification of subsymmetric basic sequences
Mar
15
comment A question on measuring weak non-compactness in $L_{1}(\mu)$ spaces
Look at the section on classical spaces in my Handbook "Basic Concepts" with Lindenstrauss. There we define three parameters $a(K)$, $b(K)$, and $c(K)$ for a subset of $L_1$ and comment that they are equal (the gist of proving this is probably contained in the books of Albiac-Kalton and Wojtaszczyk). I think it is easy to check that $c(K) = \iota(K)$ and $a(K)= \omega(K)$.
Mar
15
comment A question on measuring weak non-compactness in $L_{1}(\mu)$ spaces
Far from it. If a $C_0(K)$ space can be so written, the $X_\gamma$ are all finite dimensional and uniformly injective and $C_0(K)$ is isomorphic to $c_0(\Gamma)$.
Mar
15
comment Separable von Neumann algebra
Is it any easier to prove that an infinite dimensional $C^*$ algebra contains a self adjoint element that has infinite spectrum?
Mar
15
comment A question on measuring weak non-compactness in $L_{1}(\mu)$ spaces
I see. I did not understand your definitions at first reading.