Some approximants in operator algebras

In this paper we extend to C*-algebras and to von Neumann algebras some results on approximants that have previously been found in the context of $ \mathcal{L} $ \mathcal{L} (H) and of the von Neumann-Schatten classesC _p , 1⩽ p <∞. We obtain results concerning positive approximants, unitary and partially isometric approximants and commutator approximants; and we study paranormality. Our main tools are the Gelfand-naimark Theorem and Berntzen’s results on normal spectral approximation.