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# Exact and Approximate Operator Parallelism

Published:2014-09-26
Printed: Mar 2015
Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, Mashhad 91775, Iran
• Ali Zamani,
Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, Mashhad 91775, Iran
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## Abstract

Extending the notion of parallelism we introduce the concept of approximate parallelism in normed spaces and then substantially restrict ourselves to the setting of Hilbert space operators endowed with the operator norm. We present several characterizations of the exact and approximate operator parallelism in the algebra $\mathbb{B}(\mathscr{H})$ of bounded linear operators acting on a Hilbert space $\mathscr{H}$. Among other things, we investigate the relationship between approximate parallelism and norm of inner derivations on $\mathbb{B}(\mathscr{H})$. We also characterize the parallel elements of a $C^*$-algebra by using states. Finally we utilize the linking algebra to give some equivalence assertions regarding parallel elements in a Hilbert $C^*$-module.
 Keywords: $C^*$-algebra, approximate parallelism, operator parallelism, Hilbert $C^*$-module
 MSC Classifications: 47A30 - Norms (inequalities, more than one norm, etc.) 46L05 - General theory of $C^*$-algebras 46L08 - $C^*$-modules 47B47 - Commutators, derivations, elementary operators, etc. 15A60 - Norms of matrices, numerical range, applications of functional analysis to matrix theory [See also 65F35, 65J05]

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