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Strong Birkhoff-James orthogonality in Hilbert C*- modules

We say that two elements of a Hilbert C*-module are orthogonal if their C*-valued inner product is 0. In a Hilbert C*-module, besides this type of orthogonality, we can study all other orthogonalities defined in a general normed space. One which is most frequently used is Birkhoff–James orthogonality - if x, y are elements of a normed linear space X, then x is orthogonal to y in the BJ sense if ∥x + λy∥ ≥ ∥x∥ for all scalars λ. As we usually do in Hilbert C*-modules, we study analogous relations obtained by replacing scalars with elements of the underlying C*- algebra, or the norm with the C*- valued ”norm”. It often happens that these relations are very strong and coincide with (the first mentioned) orthogonality in a Hilbert C*- module, but not always. This leads to the notion of the strong (also called modular) BJ orthogonality which is the main topic of this talk. This is a joint work with A. Guterman, B. Kuzma, R. Rajić and S. Zhilina.

strong Birkhoff-James orthogonality, C*-algebra

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