The gap between unbounded regular operators

Hilbert $C^*$-模上无界正则算子的广义逆与极分解

Generalized inverses and polar decomposition of unbounded regular operators on Hilbert $C^*$-modules

本文研究了Hilbert $C^*$-模上无界正则算子的极分解问题，证明了其存在性等价于算子值域闭包的正交补性及无界正则广义逆的存在性，并刻画了相关$C^*$-代数为紧算子代数的条件。

Unbounded multipliers on operator spaces

We introduce unbounded multipliers on operator spaces. These multipliers generalize both, regular operators on Hilbert C*-modules and (bounded) multipliers on operator spaces.

Unbounded Operators on Hilbert $C^*$-Modules

Let $E$ and $F$ be Hilbert $C^*$-modules over a $C^*$-algebra $\CAlg{A}$. New classes of (possibly unbounded) operators $t:E\to F$ are introduced and investigated. Instead of the density of the domain $\Def(t)$ we only assume that $t$ is essentially defined, that is, $\Def(t)^\bot=\{0\}$. Then $t$ has a well-defined adjoint. We call an essentially defined operator $t$ graph regular if its graph $\Graph(t)$ is orthogonally complemented in $E\oplus F$ and orthogonally closed if $\Graph(t)^{\bot\bot}=\Graph(t)$. A theory of these operators is developed. Various characterizations of graph regular …

Locally bounded perturbations and (odd) unbounded KK-theory

A regular symmetric operator on a Hilbert module is self-adjoint whenever there exists a suitable approximate identity. We say an operator is 'locally bounded' if the composition of the operator with each element in the approximate identity is bounded. We prove that the perturbation of a regular self-adjoint operator by a locally bounded symmetric operator is again regular and self-adjoint. We use this result to show that the Kasparov class represented by an unbounded Kasparov module is stable under locally bounded perturbations. As an application, we show that we obtain a converse to the 'dou…

A local global principle for regular operators in Hilbert C*-modules

Hilbert C*-modules are the analogues of Hilbert spaces where a C*-algebra plays the role of the scalar field. With the advent of Kasparov's celebrated KK-theory they became a standard tool in the theory of operator algebras. While the elementary properties of Hilbert C*-modules can be derived basically in parallel to Hilbert space theory the lack of an analogue of the Projection Theorem soon leads to serious obstructions and difficulties. In particular the theory of unbounded operators is notoriously more complicated due to the additional axiom of regularity which is not easy to check. In this…

Differentiable absorption of Hilbert C*-modules, connections, and lifts of unbounded operators

The Kasparov absorption (or stabilization) theorem states that any countably generated Hilbert C*-module is isomorphic to a direct summand in the standard module of square summable sequences in the base C*-algebra. In this paper, this result will be generalized by incorporating a densely defined derivation on the base C*-algebra. This leads to a differentiable version of the Kasparov absorption theorem. The extra compatibility assumptions needed are minimal: It will only be required that there exists a sequence of generators with mutual inner products in the domain of the derivation. The diffe…