Characterization of numerical radius parallelism in C^*-algebras

2018/05/23 by Ali Zamani, Zamani, Ali · 1 citation
Mathematics · #46B20 #46L05 #47A12 #47A30 #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.1805.09321

openalex publication_date 2018/05/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let v(x) be the numerical radius of an element x in a C^*-algebra \mathfrakA. First, we prove several numerical radius inequalities in \mathfrakA. Particularly, we present a refinement of the triangle inequality for the numerical radius in C^*-algebras. In addition, we show that if x∈\mathfrakA, then v(x) = (1)/(2)‖x‖ if and only if ‖x‖ = ‖Re(e^iθx)‖ + ‖Im(e^iθx)‖ for all θ∈ ℝ. Among other things, we introduce a new type of parallelism in C^*-algebras based on numerical radius. More precisely, we consider elements x and y of \mathfrakA which satisfy v(x + λx) = v(x) + v(y) for some complex unit λ. We show that this relation can be characterized in terms of pure states acting on \mathfrakA.

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• A generalized Birkhoff-James orthogonality and norm parallelism in unital C^*-algebras and their characterizations

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