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General normed spaces are not equipped with an inner product. Nonetheless, there do exist several nonequivalent extensions of orthogonality from inner product spaces to general normed ones. One of the most well-known is the Birkhoff-James orthogonality: if X is a normed space and x, y elements in X, then x is Birkhoff--James orthogonal to y if the distance from x to onedimensional space spanned by y is equal to the norm of x. In a C*-algebra A we can also discuss the case when x is Birkhoff--James orthogonal to all the elements of the form ya, ain A. In this case we say that x is strongly Birkhoff-James orthogonal to y. We discuss this kind of orthogonality in commutative C*-algebras. This is a joint work with A. Guterman, B. Kuzma, R. Rajić and S. Zhilina.
strong Birkhoff-James orthogonality, Hilbert C*-modules, commutative C*-algebras
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