Quasi-free derivations on the Canonical Anticommutation Relation algebra

Abstract

An unbounded ∗-derivation δ on a C ∗-algebra U is called approximately bounded if there is an increasing sequence of full matrix subalgebras { U n } whose union is dense in the domain of U and a sequence { h n } of self-adjoint elements of U such that h n implements δ on U n for every n, and {∥ h n − Q n ( h n )∥} is a bounded sequence where Q n is the canonical conditional expectation of U onto U n . We prove that a quasi-free derivation on the Canonical Anticommutation Relation algebra is approximately bounded if the self-adjoint operator from which it arises is of finite multiplicity and bounded. We conjecture that all quasi-free derivations are approximately bounded. We also prove that a quasi-free derivation is bounded if and only if the self-adjoint operator from which it arises is of the trace class.