The main result is that the only non-trivial closed ideal in the Banach algebra L(L-p) of bounded linear operators on L-p(0,1), 1 <= p < infinity , that has a left approximate identity is the ideal of compact operators. The algebra L(L-1) has at least one non-trivial closed ideal that has a contractive right approximate identity as well as many, including the unique maximal ideal, that do not have a right approximate identity.

We survey some of the recent progress in determining the number of two-sided closed ideals in the Banach algebras of bounded linear operators on Lebesgue spaces, $L\_{p}\[0,1]$. In particular, we discuss two recent results: the first of Johnson, Pisier, and the author, showing that there are a continuum of such ideal in the case of $p=1$; the second a result of Johnson and the author, showing that in the case $1\

A Banach space X has the SHAI (surjective homomorphisms are injective) property provided that for every Banach space Y, every continuous surjective algebra homomorphism from the bounded linear operators on X onto the bounded linear operators on Y is injective. The main result gives a sufficient condition for X to have the SHAI property. The condition is satisfied for Lp(0,1) for 1<p<∞, spaces with symmetric bases that have finite cotype, and the Schatten p-spaces for 1<p<∞.

We show that there are $2^{2^{\aleph_0}}$ different closed ideals in the Banach algebra $L(L_p(0,1))$, $1

We show that there are 2^2^ℵ_0 different closed ideals in the Banach algebra L(L_p(0,1)), 1<p≠ 2<∞. This solves a problem in A. Pietsch's 1978 book "Operator Ideals". The proof is quite different from other methods of producing closed ideals in the space of bounded operators on a Banach space; in particular, the ideals are not contained in the strictly singular operators and yet do not contain projections onto subspaces that are non Hilbertian. We give a criterion for a space with an unconditional basis to have 2^2^ℵ_0 closed ideals in terms of the existence of a single operator on the space with some special asymptotic properties. We then show that for 1<q<2 the space X_q of Rosenthal, which is isomorphic to a complemented subspace of L_q(0,1), admits such an operator.

Let $${\mathsf {S}}_1$$ (the Schatten–von Neumann trace class) denote the Banach space of all compact linear operators $$T:\ell _2\rightarrow \ell _2$$ whose nuclear norm $$\Vert T\Vert _{{\mathsf {S}}_1}=\sum _{j=1}^\infty \upsigma _j(T)$$ is finite, where $$\{\upsigma _j(T)\}_{j=1}^\infty $$ are the singular values of T. We prove that for arbitrarily large $$n\in {\mathbb {N}}$$ there exists a subset $${\mathscr {C}}\subseteq {\mathsf {S}}_1$$ with $$|{\mathscr {C}}|=n$$ that cannot be embedded with bi-Lipschitz distortion O(1) into any $$n^{o(1)}$$-dimensional linear subspace of $${\mathsf {S}}_1$$. $${\mathscr {C}}$$ is not even a O(1)-Lipschitz quotient of any subset of any $$n^{o(1)}$$-dimensional linear subspace of $${\mathsf {S}}_1$$. Thus, $${\mathsf {S}}_1$$ does not admit a dimension reduction result á la Johnson and Lindenstrauss (Conference in modern analysis and probability, American Mathematical Society, Providence, 1984), which complements the work of Harrow et al. (Automata, languages and programming (ICALP 2011), Springer, Heidelberg, 2011) on the limitations of quantum dimension reduction under the assumption that the embedding into low dimensions is a quantum channel. Such a statement was previously known with $${\mathsf {S}}_1$$ replaced by the Banach space $$\ell _1$$ of absolutely summable sequences via the work of Brinkman and Charikar (J ACM 52(5):766–788, 2005). In fact, the above set $${\mathscr {C}}$$ can be taken to be the same set as the one that Brinkman and Charikar considered, viewed as a collection of diagonal matrices in $${\mathsf {S}}_1$$. The challenge is to demonstrate that $${\mathscr {C}}$$ cannot be faithfully realized in an arbitrary low-dimensional subspace of $${\mathsf {S}}_1$$, while Brinkman and Charikar obtained such an assertion only for subspaces of $${\mathsf {S}}_1$$ that consist of diagonal operators (i.e., subspaces of $$\ell _1$$). We establish this by proving that the Markov 2-convexity constant of any finite dimensional linear subspace X of $${\mathsf {S}}_1$$ is at most a universal constant multiple of $$\sqrt{\log \dim (X)}$$.

The main result is that there are infinitely many; in fact, a continuum; of closed ideals in the Banach algebra $L(L_1)$ of bounded linear operators on $L_1(0,1)$. This answers a question from A. Pietsch's 1978 book Operator Ideals. The proof also shows that $L(C[0,1])$ contains a continuum of closed ideals. Finally, a duality argument yields that $L(\ell_\infty)$ has a continuum of closed ideals.

We present an example of a function $f$ from $\{-1,1\}^n$ to the unit sphere in $\mathbb{C}$ with influence bounded by $1$ and entropy of $|\hat f|^2$ larger than $\frac12\log n$. We also present an example of a function $f$ from $\{-1,1\}^n$ to $\mathbb{R}$ with $L_2$ norm $1$, $L_\infty$ norm bounded by $\sqrt{2}$, influence bounded by $1$ and entropy of $\hat f^2$ larger than $\frac12\log n$.

We show that a subspace of l(infinity)(N) of dimension n > (logN log log N)(2) contains 2-isomorphic copies of l(infinity)(k) where k tends to infinity with n/(logN log logN)(2). More precisely, for every eta > 0, we show that any subspace of l(infinity)(N) of dimension n contains a subspace of dimension m = c(eta)root n/(logN log logN) of distance at most 1 + eta from l(infinity)(m).

The main result is that there are infinitely many; in fact, a continuum; of closed (twosided) ideals in the Banach algebra L(L1) of bounded linear operators on L1(0, 1). This answers a question from A. Pietsch’s 1978 book “Operator Ideals”. The proof also shows that L(C[0, 1]) contains a continuum of closed ideals. Finally, a duality argument yields that L(l∞) has a continuum of closed ideals.

We show that a subspace of of l∞ of dimension n > (logN log logN) 2 contains 2-isomorphic copies of l∞ where k tends to infinity with n/(logN log logN)2. More precisely, for every η > 0, we show that any subspace of l∞ of dimension n contains a subspace of dimension m = c(η) √ n/(logN log logN) of distance at most 1 + η from l∞.

For every p is an element of (0, infinity) we associate metric space ( X, d(x)) a numerical invariant x(p) (X) is an element of [0, infinity] such that if x(p) (X) < infinity, and a metric space (Y, d(Y) ) admits a bi-Lipschitz embedding into X then also x(p) (Y) < infinity. We prove that if p, q is an element of (2, infinity) satisfy q < p then x(p) (L-p) < infinity yet x(p) (L-q) = infinity,. Thus, our new bi-Lipschitz invariant certifies that L, does not admit a hi -Lipschitz embedding into 1, when 2 < q < p < infinity,. This completes the long-standing search for hi -Lipschitz invariants that serve as an obstruction to the embeddahility of L spaces into each other, the previously understood cases of which were metric notions of type and cotype, which however fail to certify the nonembeddability of L-q, into L1, when 2 < q < p < infinity Among the consequences of our results are new quantitative restrictions on the bi-Lipschitz embeddability into Lp of snowflakes of L, and integer grids in l(q)(n), for 2 t, q < p < infinity.). Asa byproduct of our investigations, we also obtain results on the geometry of the Schatten p trace class Sp that are new even in the linear setting.

For $n\in \mathbb{N}$ consider the $n$-dimensional hypercube as equal to the vector space $\mathbb{F}_2^n$, where $\mathbb{F}_2$ is the field of size two. Endow $\mathbb{F}_2^n$ with the Hamming metric, i.e., with the metric induced by the $\ell_1^n$ norm when one identifies $\mathbb{F}_2^n$ with $\{0,1\}^n\subseteq \mathbb{R}^n$. Denote by $\ell_2^n(\mathbb{F}_2^n)$ the $n$-fold Pythagorean product of $\mathbb{F}_2^n$, i.e., the space of all $x=(x_1,\ldots,x_n)\in \prod_{j=1}^n \mathbb{F}_2^n$, equipped with the metric $$ \forall\, x,y\in \prod_{j=1}^n \mathbb{F}_2^n,\qquad d_{\ell_2^n(\mathbb{F}_2^n)}(x,y)= \sqrt{ \|x_1-y_1\|_1^2+\ldots+\|x_n-y_n\|_1^2}. $$ It is shown here that the bi-Lipschitz distortion of any embedding of $\ell_2^n(\mathbb{F}_2^n)$ into $L_1$ is at least a constant multiple of $\sqrt{n}$. This is achieved through the following new bi-Lipschitz invariant, which is a metric version of (a slight variant of) a linear inequality of Kwapie{\'n} and Sch\utt (1989). Letting $\{e_{jk}\}_{j,k\in \{1,\ldots,n\}}$ denote the standard basis of the space of all $n$ by $n$ matrices $M_n(\mathbb{F}_2)$, say that a metric space $(X,d_X)$ is a KS space if there exists $C=C(X)>0$ such that for every $n\in 2\mathbb{N}$, every mapping $f:M_n(\mathbb{F}_2)\to X$ satisfies \begin{equation*}\label{eq:metric KS abstract} \frac{1}{n}\sum_{j=1}^n\mathbb{E}\left[d_X\Big(f\Big(x+\sum_{k=1}^ne_{jk}\Big),f(x)\Big)\right]\le C \mathbb{E}\left[d_X\Big(f\Big(x+\sum_{j=1}^ne_{jk_j}\Big),f(x)\Big)\right], \end{equation*} where the expectations above are with respect to $x\in M_n(\mathbb{F}_2)$ and $k=(k_1,\ldots,k_n)\in \{1,\ldots,n\}^n$ chosen uniformly at random. It is shown here that $L_1$ is a KS space (with $C= 2e^2/(e^2-1)$, which is best possible), implying the above nonembeddability statement. Links to the Ribe program are discussed, as well as related open problems.

The objectives of this note are to correct a common error and to clarify the connection between the Gini terminology as used in the economic literature and the one used in the diagnostic and classification literature. More specifically, the connection between the area under the receiver operating characteristic (ROC) curve, which is frequently used in the diagnosis and classification literature, and the Gini terminology, which is mainly used in the economic literature, is clarified. It is shown that the area under the ROC curve is related to the covariance between the two vectors $$Y=\{y_i\}_{i=1}^{n_0}$$ and $$\{i/{n_0}\}_{i=1}^{n_0}$$. Here $$y_i$$ is the number of items classified to group 1 lying between the $$(i-1)\mathrm{th}$$ and the $$i\mathrm{th}$$ items classified to group 0, and $$n_0$$ is the number of items in group 0.

Let $\lambda$ be an infinite cardinal number and let $\ell_\infty^c(\lambda)$ denote the subspace of $\ell_\infty(\lambda)$ consisting of all functions that assume at most countably many non-zero values. We classify all infinite dimensional complemented subspaces of $\ell_\infty^c(\lambda)$, proving that they are isomorphic to $\ell_\infty^c(\kappa)$ for some cardinal number $\kappa$. Then we show that the Banach algebra of all bounded linear operators on $\ell_\infty^c(\lambda)$ or $\ell_\infty(\lambda)$ has the unique maximal ideal consisting of operators through which the identity operator does not factor. Using similar techniques, we obtain an alternative to Daws' approach description of the lattice of all closed ideals of $\mathscr{B}(X)$, where $X = c_0(\lambda)$ or $X=\ell_p(\lambda)$ for some $p\in [1,\infty)$, and we classify the closed ideals of $\mathscr{B}(\ell_\infty^c(\lambda))$ that contains the ideal of weakly compact operators.

The paper contains three results, the common feature of which is that they deal with the Schatten $p$ class. The first is a presentation of a new complemented subspace of $C_p$ in the reflexive range (and $p\not= 2$). This construction answers a question of Arazy and Lindestrauss from 1975. The second result relates to tight embeddings of finite dimensional subspaces of $C_p$ in $C_p^n$ with small $n$ and shows that $\ell_p^k$ nicely embeds into $C_p^n$ only if $n$ is at least proportional to $k$ (and then of course the dimension of $C_p^n$ is at least of order $k^2$). The third result concerns single element of $C_p^n$ and shows that for $p>2$ any $n\times n$ matrix of $C_p$ norm one and zero diagonal admits, for every $\varepsilon>0$, a $k$-paving of $C_p$ norm at most $\varepsilon$ with $k$ depending on $\varepsilon$ and $p$ only.

We construct a Schauder basis for L 1 consisting of non-negative functions and investigate unconditionally basic and quasibasic sequences of non-negative functions in L p , 1 ≤ p < ∞.

Let A be an m x m complex matrix with zero trace. Then there are m x m matrices B and C such that A = [B, C] and parallel to B parallel to parallel to C parallel to(2) <= (log m + Omicron(1))(1/2) parallel to A parallel to (2) where parallel to D parallel to is the norm of D as an operator on l(2)(m) and parallel to D parallel to(2) is the Hilbert-Schmidt norm of D. Moreover, the matrix B can be taken to be normal. Conversely, there is a zero trace m x m matrix A such that whenever A = [B, C], parallel to B parallel to parallel to C parallel to(2) >= vertical bar log m - Omicron(1) |(1/ 2) parallel to A parallel to(2) for some absolute constant c > 0.

Joram's PhD dealt with extensions of linear operators between Banach spaces, leading also to the study of preduals of L 1 spaces.Some of his other groundbreaking research results include a study with A. Pełczyński of Grothendieck's work in Banach space theory and applications thereof,