The principle of local reflexivity for operator ideals and its implications

Operators with extension property and the principle of local reflexivity

Given an arbitrary $p$-Banach ideal $(0 < p \leq 1)$, we ask for geometrical properties of this ideal which are sufficient (and necessary) to allow a transfer of the principle of local reflexivity to this operator class.

A Short Note on some open problems in the geometry of operator ideals

We list and discuss the background of some open problems, regarding the principle of local reflexivity for maximal Banach ideals.

Extension of Finite Rank Operators and Local Structures in Operator Ideals

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…

Multiplicative maps on ideals of operators which are local automorphisms

We present the following reflexivity-like result concerning the automorphism group of the $C^*$-algebra B(H), H being a separable Hilbert space. Let $ϕ:B(H)\to B(H)$ be a multiplicative map (no linearity or continuity is assumed) which can be approximated at every point by automorphisms of B(H) (these automorphisms may, of course, depend on the point) in the operator norm. Then $ϕ$ is an automorphism of the algebra B(H).

Classification of implication-closed ideals in certain rings of jets

For a set $E\subset\mathbb{R}^n$ that contains the origin we consider $I^m(E)$ -- the set of all $m^{\text{th}}$ degree Taylor approximations (at the origin) of $C^m$ functions on $\mathbb{R}^n$ that vanish on $E$. This set is a proper ideal in $\mathcal{P}^m(\mathbb{R}^n)$ -- the ring of all $m^{\text{th}}$ degree Taylor approximations of $C^m$ functions on $\mathbb{R}^n$. In [FS] we introduced the notion of a \textit{closed} ideal in $\mathcal{P}^m(\mathbb{R}^n)$, and proved that any ideal of the form $I^m(E)$ is closed. In this paper we classify (up to a natural equivalence relation) all cl…

关于Banach代数中稠密理想上内导出的扩展

On the extension of inner derivations from dense ideals in Banach algebras

本文探讨了Banach代数中稠密理想上内导出的扩展问题，通过构造紧算子代数和有限秩算子理想，证明了所有导出到有限秩算子的理想都是内导出，但紧算子代数上存在外导出。