Characterizations of operator Birkhoff-James orthogonality

Orthogonality in Banach Spaces via projective tensor product

Let $X$ be a complex Banach space and $x,y\in X$. By definition, we say that $x$ is Birkhoff-James orthogonal to $y$ if $ \|x+λy\|_{X} \geq \|x\|_{X}$ for all $λ\in \mathbb{C}$. We prove that $x$ is Birkhoff-James orthogonal to $y$ if and only if there exists a semi-inner product $φ$ on $X$ such that $\|φ\| = 1$, $φ(x,x)=\|x\|^2$ and $φ(x,y)=0$. A similar result holds for $C^*$-algebras. A key point in our approach to orthogonality is the representations of bounded bilinear maps via projective tensor product spaces.

通过正交性对阿贝尔有限维$C^*$-代数的分类

Classification of abelian finite-dimensional $C^*$-algebras by orthogonality

本文研究有限维交换$C^*$-代数的Birkhoff-James同构分类问题，利用左对称元及其正交补刻画极小理想结构，证明维数大于1时该同构蕴含底域相同且代数等距$*$-同构，并给出提取域型极小理想的方法。

Operator equalities and Characterizations of Orthogonality in Pre-Hilbert $C^*$-Modules

In the first part of the paper, we use states on $C^*$-algebras in order to establish some equivalent statements to equality in the triangle inequality, as well as to the parallelogram identity for elements of a pre-Hilbert $C^*$-module. We also characterize the equality case in the triangle inequality for adjointable operators on a Hilbert $C^*$-module. Then we give certain necessary and sufficient conditions to the Pythagoras identity for two vectors in a pre-Hilbert $C^*$-module under the assumption that their inner product has negative real part. We introduce the concept of Pythagoras orth…

Norm-parallelism in the geometry of Hilbert $C^*$-modules

Utilizing the Birkhoff--James orthogonality, we present some characterizations of the norm-parallelism for elements of $\mathbb{B}(\mathscr{H})$ defined on a finite dimensional Hilbert space, elements of a Hilbert $C^*$-module over the $C^*$-algebra of compact operators and elements of an arbitrary $C^*$-algebra. We also consider the characterization of norm parallelism problem for operators on a finite dimensional Hilbert space when the operator norm is replaced by the Schatten $p$-norm. Some applications and generalizations are discussed for certain elements of a Hilbert $C^*$-module.