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Orthograph related to mutual strong Birkhoff-James orthogonality in \(C^*\)-algebras. (English) Zbl 1466.46043

Let \(A\) be a \(C^*\)-algebra, \(a,b\in A\). We write \(a\perp^s_{BJ} b\) (\(a\) is strongly Birkhoff-James orthogonal to \(b\)) if \(\|a+bc\|\geq \|a\|\) for any \(c\in A\). \(a\) and \(b\) are mutually strongly Birkhoff-James orthogonal if \(a\perp^s_{BJ}b\) and \(b\perp^s_{BJ}a\). The paper deals with the orthograph of \(A\). Its vertices are one-dimensional subspaces of \(A\), and a pair of vertices \([a]\), \([b]\) is connected by an edge (of length one) if \(a\) and \(b\) are mutually strongly Birkhoff-James orthogonal.
If \(A=\mathbb B(H)\) is the algebra of all bounded operators on an infinite dimensional Hilbert space, then it is shown that the set of not right-invertible elements of \(A\) is a connected component of the orthograph whose diameter is 3, while each right-invertible element is an isolated point of the orthograph.
If \(A=C(X)\) is the algebra of continuous functions on a compact Hausdorff space \(X\), then the isolated points of the orthograph are exactly the invertible elements of \(A\), and the non-invertible elements of \(A\) form a connected component of the orthograph with diameter 3 if at least one point of \(X\) has a countable local basis, otherwise its diameter is 2.

MSC:

46L05 General theory of \(C^*\)-algebras
46B20 Geometry and structure of normed linear spaces
05C12 Distance in graphs
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
Full Text: DOI

References:

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