# Derivations, local and 2-local derivations on some algebras of operators on Hilbert C*-modules

## TL;DR

This paper investigates derivations and 2-local derivations on algebras of bounded and adjointable operators over Hilbert C*-modules, proving their linearity, continuity, and inner nature under certain conditions.

## Contribution

It establishes that derivations on these operator algebras are inner and that 2-local derivations are actual derivations, extending understanding of their structure.

## Key findings

- Derivations are $ ext{A}$-linear, continuous, and inner.
- 2-local derivations are actual derivations.
- Conditions for local derivations to be derivations are identified.

## Abstract

For a commutative C*-algebra $\mathcal A$ with unit $e$ and a Hilbert~$\mathcal A$-module $\mathcal M$, denote by End$_{\mathcal A}(\mathcal M)$ the algebra of all bounded $\mathcal A$-linear mappings on $\mathcal M$, and by End$^*_{\mathcal A}(\mathcal M)$ the algebra of all adjointable mappings on $\mathcal M$. We prove that if $\mathcal M$ is full, then each derivation on End$_{\mathcal A}(\mathcal M)$ is $\mathcal A$-linear, continuous, and inner, and each 2-local derivation on End$_{\mathcal A}(\mathcal M)$ or End$^{*}_{\mathcal A}(\mathcal M)$ is a derivation. If there exist $x_0$ in $\mathcal M$ and $f_0$ in $\mathcal M^{'}$, such that $f_0(x_0)=e$, where $\mathcal M^{'}$ denotes the set of all bounded $\mathcal A$-linear mappings from $\mathcal M$ to $\mathcal A$, then each $\mathcal A$-linear local derivation on End$_{\mathcal A}(\mathcal M)$ is a derivation.