Approximately invertible elements in non-unital normed algebras

Esmeral, Kevin; Feichtinger, Hans G.; Hutník, Ondrej; Maximenko, Egor A.

doi:10.1016/j.jmaa.2022.126986

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Isolated Vertices2

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2023

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“…We need the notion of approximate invertibility, introduced in [8] in the following way Definition 2.1. Let A be a C * -algebra and let a ∈ A.…”

Section: Isolated Verticesmentioning

confidence: 99%

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Isolated vertices and diameter of the $BJ$-orthograph in $C^*$-algebras

Kečkić¹,

Stefanović²

2023

Preprint

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We give necessary and sufficient condition that an element of an arbitrary C * -algebra is an isolated vertex of the orthograph related to the mutual strong Birkhoff-James orthogonality. Also, we prove that for all C *algebras except C, C ⊕ C and M 2 (C) all non isolated points make a single connected component of the orthograph which diameter is less than or equal to 4, i.e. any two non isolated points can be connected by a path with at most 4 edges. Some related results are given. Definition 1.2. Let A be a C * -algebra and let a, b ∈ A. a) We say that a is strong BJ-orthogonal to b, and denote a ⊥ S b if for any c ∈ A there holds (1.2) a + bc ≥ a . b) We say that a and b are mutually strong BJ-orthogonal to each other, and denote a ⊥ ⊥ S b if a ⊥ S b and b ⊥ S a. Remark 1.1. The part a) of the previous definition is from [7], and part b) from [6].

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“…We need the notion of approximate invertibility, introduced in [8] in the following way Definition 2.1. Let A be a C * -algebra and let a ∈ A.…”

Section: Isolated Verticesmentioning

confidence: 99%

“…We say that a is approximately right invertible if there is a net {b i } i∈I such that bb i is approximate unit for A. Some results given in [8] Proof. This is essentially [8,Theorem 3.6].…”

Section: Isolated Verticesmentioning

confidence: 99%