Normed Ideal Perturbation of Irreducible Operators in Semifinite Von Neumann Factors

An interesting result proved by Halmos in Hal (Michigan Mathematical Journal, 15, 215–223 (1968) is that the set of irreducible operators dense $${\mathcal {B}}({\mathcal {H}})$$ sense Hilbert-Schmidt approximation. In a von Neumann algebra {M}}$$ with separable predual, an operator $$a\in {\mathcal said to be if $$W^*(a)$$ subfactor , i.e., $$W^*(a)'\cap {M}}={{\mathbb {C}}} \cdot I$$ . Let $$\Phi (\cdot )$$ $$\Vert \Vert $$ -dominating, unitarily invariant norm (see Definition 2.1). Many well-known norms are this type such as and $$\max \{\Vert _p\}$$ -norm (for each $$p>1$$ ). Theorem 3.1, we prove every semifinite factor satisfies natural condition introduced (1.1), then particular, ) naturally satisfy (1.1). This can viewed (stronger) analogue theorem (1968), different techniques developed semifinite, properly infinite factors predual. It ask whether (1.1) necessary for 3.1. We not normal operators. 4.1, develop another method sum arbitrarily small perturbation. Particularly, trace class _1$$ on