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Best approximations, distance formulas and orthogonality in C*-algebras

TL;DR: In this article, a characterization for a best approximation to an element of a Hilbert module in a subspace of a unital $C^*-algebra has been obtained, where the subspace is a Hilbert subspace.
Abstract: For a unital $C^*$-algebra $\mathcal A$ and a subspace $\mathcal B$ of $\mathcal A$, a characterization for a best approximation to an element of $\mathcal A$ in $\mathcal B$ is obtained. As an application, a formula for the distance of an element of $\mathcal A$ from $\mathcal B$ has been obtained, when a best approximation of that element to $\mathcal B$ exists. Further, a characterization for Birkhoff-James orthogonality of an element of a Hilbert $C^*$-module to a subspace is obtained.
Citations
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Journal ArticleDOI
TL;DR: In this article, the Gateaux derivative of the C ⁎ -algebra norm is characterized under the assumption that dist(A, K (H, K, K ) ) is a subspace.

10 citations

Book ChapterDOI
01 Jan 2021
TL;DR: In this paper, the necessary and sufficient conditions for Birkhoff-James orthogonality in Banach spaces are discussed and their applications in studying the geometry of normed spaces are given.
Abstract: In the last few decades, the concept of Birkhoff–James orthogonality has been used in several applications. In this survey article, the results known on the necessary and sufficient conditions for Birkhoff–James orthogonality in certain Banach spaces are mentioned. Their applications in studying the geometry of normed spaces are given. The connections between this concept of orthogonality, and the Gateaux derivative and the subdifferential set of the norm function are provided. Several interesting distance formulas can be obtained using the characterizations of Birkhoff–James orthogonality, which are also mentioned. In the end, some new results are obtained.

6 citations

Posted Content
TL;DR: In this paper, an expression for Gateaux derivative of the $C^*$-algebra norm is given for spaces with subdifferential sets, smooth points and Birkhoff-James orthogonality.
Abstract: We find an expression for Gateaux derivative of the $C^*$-algebra norm. This gives us alternative proofs or generalizations of various known results on the closely related notions of subdifferential sets, smooth points and Birkhoff-James orthogonality for spaces $\mathscr B(\mathcal H)$ and $C_b(\Omega)$. We also obtain an expression for subdifferential sets of the norm function at $A\in\mathscr B(\mathcal H)$ and a characterization of orthogonality of an operator $A\in\mathscr B(\mathcal H, \mathcal K)$ to a subspace, under the condition $dist(A, \mathscr K(\mathcal H))< \|A\|$ and $dist(A, \mathscr K(\mathcal H, \mathcal K))< \|A\|$ respectively.
References
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Book
01 Jan 1985
TL;DR: In this article, an introductory text in functional analysis aimed at the graduate student with a firm background in integration and measure theory is presented, which helps the student to develop an intuitive feel for the subject.
Abstract: This book is an introductory text in functional analysis, aimed at the graduate student with a firm background in integration and measure theory. Unlike many modern treatments, this book begins with the particular and works its way to the more general, helping the student to develop an intuitive feel for the subject. For example, the author introduces the concept of a Banach space only after having introduced Hilbert spaces, and discussing their properties. The student will also appreciate the large number of examples and exercises which have been included.

2,519 citations

Book
01 Nov 1970
TL;DR: In this paper, the authors propose an approximation of sous-espace lineaire de dimension finie, which is a lineaire lineaire of the dimension of the element d'ensemble.
Abstract: espace lineaire norme # espace metrique # meilleure approximation # sous-espace lineaire # sous-espace lineaire de dimension finie # sous-espace lineaire ferme de codimension finie # element d'ensemble # element d'ensemble non-lineaire

756 citations

Book
08 Dec 2005
TL;DR: In this article, the authors present a model for operators on Hilbert Space, including C*-Algebras, Von Neumann Algebra, and K-Theory and Finiteness.
Abstract: Operators on Hilbert Space.- C*-Algebras.- Von Neumann Algebras.- Further Structure.- K-Theory and Finiteness.

632 citations

Book
01 Jan 1982

149 citations

Journal ArticleDOI
TL;DR: In this article, necessary and sufficient conditions for orthogonality between two matrices are given, such that A and B are matrices such that ||A + zB|| ⩾ ||A|| for all complex numbers z.

134 citations