We give necessary and sufficient condition that an element of arbitrary C⁎-algebra is isolated vertex the orthograph related to mutual strong Birkhoff-James orthogonality. Also, we prove for all C⁎-algebras except C,C⊕C M2(C) non points make a single connected component which diameter less than or equal 4, i.e. any two can be by path with at most 4 edges. Some results are given.

We find an expression for the Gateaux derivative of C⁎-algebra norm. Using this, we obtain a characterization orthogonality operator A∈B(H,K) to subspace, under assumption dist(A,K(H,K))<‖A‖. subdifferential set norm function at A∈B(H) when dist(A,K(H))<‖A‖. also give new proofs known results on closely related notions smooth points and Birkhoff-James spaces B(H) Cb(Ω), respectively.

Introducing the concept of the normalized duality mapping on normed linear space and normed algebra, we extend the usual definitions of the numerical range from one operator to two operators. In this note we study the convexity of these types of numerical ranges in normed algebras and linear spaces. We establish some Birkhoff-James orthogonality results in terms of the algebra numerical range V...

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

Let (X,⟨⋅,⋅⟩) be a Hilbert C∗-module over C∗-algebra A and let S(A) the set of states on A. In this paper, we first compute norm derivative for nonzero elements x y X as follows: limt→0+‖x+ty‖−‖x‖t=1‖x‖max{Reφ(⟨x,y⟩):φ∈S(A),φ(⟨x,x⟩)=‖x‖2}.We then apply it to characterize different concepts orthogonality in X. particular, present simpler proof classical characterization Birkhoff–James C∗-modules...