In addition to Pisier's counterexample of a non-accessible maximal Banach ideal, we will give a large class of maximal Banach ideals which {\it{are accessible}}. The first step is implied by the observation that a "good behaviour" of trace duality, which is canonically induced by conjugate operator ideals can be extended to adjoint Banach ideals, if and only if these adjoint ideals satisfy an accessibility condition (theorem 3.1). This observation leads in a natural way to a characterization of accessible injective Banach ideals, where we also recognize the appearance of the ideal of {\it{absolutely summing operators}} (prop. 4.1). By the famous {\it{Grothendieck inequality}}, every operator from $L_1$ to a Hilbert space is absolutely summing, and therefore our search for such ideals will be directed towards Hilbert space factorization - via an operator version of Grothendieck's inequality (lemma 4.2). As a consequence,we obtain a class of injective ideals, which are "quasi-accessible", and with the help of {\it{tensor stability}}, we improve the corresponding norm inequalities, to get accessibility (theorem 4.1 and 4.2). In the last chapter of this paper we give applications, which are implied by a non-trivial link of the above mentioned considerations to normed products of operator ideals.
After sketching the basic theory of injective ideals of homogeneous polynomials, we characterize injective polynomial ideals by means of a domination property and applications of this characterization to some classical operator ideals and to composition polynomial ideals are provided.
We define an $AW^*$-TRO as an off-diagonal corner of an $AW^*$-algebra, and show that the unit ball of an $AW^*$-TRO has an extreme point. In particular, the unit ball of an injective operator space has an extreme point, which answers a question raised in the author's previous work [Journal of Operator Theory, 76(2) (2016), 219-248] affirmatively. We also show that an $AW^*$-TRO (respectively, an injective operator space) has an ideal decomposition, that is, it can be decomposed into the direct sum of a left ideal, a right ideal, and a two-sided ideal in an $AW^*$-algebra (respectively, an inj…
We provide a short characterization of $p$-asymptotic uniform smoothability and asymptotic uniform flatenability of operators and of Banach spaces. We use these characterizations to show that many asymptotic uniform smoothness properties pass to injective tensor products of operators and of Banach spaces. In particular, we prove that the injective tensor product of two asymptotically uniformly smooth Banach spaces is asymptotically uniformly smooth. We prove that for $1<p<\infty$, the class of $p$-asymptotically uniformly smoothable operators can be endowed with an ideal norm making this class…
We show that the numerical index of any operator ideal is less than or equal to the minimum of the numerical indices of the domain and the range. Further, we show that the numerical index of the ideal of compact operators or the ideal of weakly compact operators is less than or equal to the numerical index of the dual of the domain, and this result provides interesting examples. We also show that the numerical index of a projective or injective tensor product of Banach spaces is less than or equal to the numerical index of any of the factors. Finally, we show that if a projective tensor produc…
We study automatic injectivity of surjective algebra homomorphisms from $\mathscr{B}(X)$, the algebra of (bounded, linear) operators on $X$, to $\mathscr{B}(Y)$, where $X$ is one of the following \emph{long} sequence spaces: $c_0(λ)$, $\ell_{\infty}^c(λ)$, and $\ell_p(λ)$ ($1 \leqslant p < \infty$) and $Y$ is arbitrary. \textit{En route} to the proof that these spaces do indeed enjoy such a property, we classify two-sided ideals of the algebra of operators of any of the aforementioned Banach spaces that are closed with respect to the `sequential strong operator topology'.
For $α$ an ordinal, we investigate the class $\mathscr{SZ}_α$ consisting of all operators whose Szlenk index is an ordinal not exceeding $ω^α$. Our main result is that $\mathscr{SZ}_α$ is a closed, injective, surjective operator ideal for each $α$. We also study the relationship between the classes $\mathscr{SZ}_α$ and several well-known closed operator ideals.
For an operator ideal $\mathcal A$, we study the composition operator ideals ${\mathcal A}\circ{\mathcal K}$, ${\mathcal K}\circ{\mathcal A}$ and ${\mathcal K}\circ{\mathcal A}\circ{\mathcal K}$, where $\mathcal K$ is the ideal of compact operators. We introduce a notion of an $\mathcal A$-approximation property on a Banach space and characterise it in terms of the density of finite rank operators in ${\mathcal A}\circ{\mathcal K}$ and ${\mathcal K}\circ{\mathcal A}$. We propose the notions of $\ell_{\infty}$-extension and $\ell_{1}$-lifting properties for an operator ideal $\mathcal A$ and st…
Given two quasi-Banach ideals \oid{A}{}{} and \oid{B}{}{} we investigate the regular hull of their composition - $(\oid{A}{}{} \circ \oid{B}{}{})^{reg}$. In concrete situations this regular hull appears more often than the composition itself. As a first example we obtain a description for the regular hull of the nuclear operators which is a "reflected" Grothendieck representation:\\ $\oid{N}{}{reg} \stackrel{1}{=} \oid{I}{}{} \circ \oid{W}{}{}$ (theorem 2.1). Further we recognize that the class of such ideals leads to interesting relations concerning the question of the accessibility of (injec…
We study tensor norms that destroy unconditionality in the following sense: for every Banach space $E$ with unconditional basis, the $n$-fold tensor product of $E$ (with the corresponding tensor norm) does not have unconditional basis. We establish an easy criterion to check weather a tensor norm destroys unconditionality or not. Using this test we get that all injective and projective tensor norms different from $\varepsilon$ and $π$ destroy unconditionality, both in full and symmetric tensor products. We present applications to polynomial ideals: we show that many usual polynomial ideals nev…
In this paper, we investigate the notion $ϕ$-injectivity for Banach $A$-modules, where $ϕ$ is a character on $A.$ We obtain some hereditary properties of $ϕ$-injectivity for certain classes of Banach modules related to closed ideals. These results allow us to study $ϕ$-injectivity of certain Banach $A$-modules in commutative case, specially $\ell^{1}$-semilattice algebras. As an application, we give an example of a non-injective Banach module which is $ϕ$-injective for each character $ϕ.$
We treat the general theory of nonlinear ideals and extend as many notions as possible from the linear theory to the nonlinear theory. We define nonlinear ideals with special properties which associate new non-linear ideals to given ones and establish several properties and characterizations of them. Building upon the results of U. Matter we define a Lipschitz interpolative nonlinear ideal procedure between metric spaces and Banach spaces and establish this class of Lipschitz operators is an injective Banach nonlinear ideal and show several standard basic properties for such class. Extending t…
We introduce a concept of bilinear ideal of jointly completely bounded mappings between operator spaces. In particular, we study the bilinear ideals $\mathcal{N}$ of completely nuclear, $\mathcal{I }$ of completely integral, $\mathcal{E}$ of completely extendible bilinear mappings, $\mathcal{MB}$ multiplicatively bounded and its symmetrization $\mathcal{SMB}$. We prove some basic properties of them, one of which is the fact that $\mathcal{I}$ is naturally identified with the ideal of (linear) completely integral mappings on the injective operator space tensor product.
We study the space of $p$-compact operators $\mathcal K_p$, using the theory of tensor norms and operator ideals. We prove that $\mathcal K_p$ is associated to $/d_p$, the left injective associate of the Chevet-Saphar tensor norm $d_p$ (which is equal to $g_{p'}'$). This allows us to relate the theory of $p$-summing operators with that of $p$-compact operators. With the results known for the former class and appropriate hypothesis on $E$ and $F$ we prove that $\mathcal K_p(E;F)$ is equal to $\mathcal K_q(E;F)$ for a wide range of values of $p$ and $q$, and show that our results are sharp. We a…
We study distinguished objects in the category $\mathcal{BL}$ of Banach lattices and lattice homomorphisms. The free Banach lattice construction introduced by de Pagter and Wickstead generates push-outs, and combining this with an old result of Kellerer on marginal measures, the amalgamation property of Banach lattices is established. This will be the key tool to prove that $L_1([0,1]^{\mathfrak{c}})$ is separably $\mathcal{BL}$-injective, as well as to give more abstract examples of Banach lattices of universal disposition for separable sublattices. Finally, an analysis of the ideals on $C(Δ,…
We examine rigidity phenomena for representations of amenable operator algebras which have an ideal of compact operators. We establish that a generalized version of Kadison's conjecture on completely bounded homomorphisms holds for the algebra if the associated quotient is abelian. We also prove that injective completely bounded representations of the algebra are similar to complete isometries. The main motivating example for these investigations is the recent construction of Choi, Farah and Ozawa of an amenable operator algebra that is not similar to a $C^*$-algebra, and we show how it fits i…
We develop general techniques and present an approach to solve the problem of constructing a maximal Banach ideal $({\frak A},{\bf A)}$ which does not satisfy a transfer of the norm estimation in the principle of local reflexivity to its norm ${\bf A}$. This approach leads us to the investigation of product operator ideals containing ${\frak L}_2$ (the collection of all Hilbertian operators) as a factor. Using the local properties of such operator ideals -- which are typical examples of ideals with property (I) and property (S) --, trace duality and an extension of suitable finite rank operato…
For each ordinal $ξ$ and each $1<p<\infty$, we offer a natural, ismorphic characterization of those spaces and operators which admit an equivalent $ξ$-$p$-asymptotically uniformly smooth norm. We also introduce the notion of $ξ$-asymptotically uniformly flat norms and provide an isomorphic characterization of those spaces and operators which admit an equivalent $ξ$-asymptotically uniformly flat norm. Given a compact, Hausdorff space $K$, we prove an optimal renormong theorem regarding the $ξ$-asymptotic smoothness of $C(K)$ in terms of the Cantor-Bendixson index of $K$. We also prove that for …
