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Diameter of the $BJ$-orthograph in finite dimensional $C^{*}$-algebras

Srdjan Stefanović

Abstract

We determine the exact diameter of the orthograph related to mutual strong Birkhoff-James orthogonality in arbitrary finite dimensional $C^{*}$-algebra. Additionally, we will estimate the distance between vertices in an arbitrary $C^{*}$-algebra that can be represented as a direct sum.

Diameter of the $BJ$-orthograph in finite dimensional $C^{*}$-algebras

Abstract

We determine the exact diameter of the orthograph related to mutual strong Birkhoff-James orthogonality in arbitrary finite dimensional -algebra. Additionally, we will estimate the distance between vertices in an arbitrary -algebra that can be represented as a direct sum.
Paper Structure (13 sections, 20 theorems, 27 equations)

This paper contains 13 sections, 20 theorems, 27 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Distance between vertices in direct sum $C^{*}$-algebras
  3. Diameter of finite dimensional $C^*$-algebras
  4. Case of $\mathbb{C}^{k}, k\in\mathbb{N}$
  5. Case of $\mathbb{C}\oplus\mathbb{C}$
  6. Case of $\mathbb{C}^{k}, k\geqslant3$
  7. Cases with two summands
  8. Case of $M_n(\mathbb{C})\oplus M_k(\mathbb{C}),n,k\geqslant2$
  9. Case of $\mathbb{C}\oplus M_2(\mathbb{C})$
  10. Case of $\mathbb{C}\oplus M_3(\mathbb{C})$
  11. Case of $\mathbb{C}\oplus M_n(\mathbb{C}), n\geqslant4$
  12. Case with three or more summands
  13. Future study

Key Result

Theorem 1.1

Let A be a finite dimensional $C^{*}$-algebra. Then A is isomorphic to a direct sum of matrix algebras, i.e., there exist positive integers $n_1, n_2,\dots, n_k$ such that

Theorems & Definitions (39)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.1
  • Definition 1.3
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 1.4
  • Example 1.1
  • Theorem 2.1
  • ...and 29 more