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We study the subset in a unital C∗-algebra composed of elements a such that aa† − aa† is invertible, where a† denotes the Moore–Penrose inverse of a. A distinguished subset of this set is also investigated. Furthermore we study sequences of elements belonging to the aforementioned subsets.

Invertibility of the commutator of an element in a C*-algebra and its Moore–Penrose inverse

Invertibility of the commutator of an element in a C^*-algebra and its Moore–Penrose inverse