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Abstract:
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The subject of the dissertation is the investigation of the relation of
strong BJ orthogonality in C∗-algebras. For two elements a and b of C∗-algebra
A, we say that a is strong BJ orthogonal to b, if for all c ∈ A holds ‖a + bc‖ ⩾ ‖a‖
and we write a ⊥S b. If it is also true that b ⊥S a, then we say that a and b are
mutual strong BJ orthogonal and write a ⊥⊥S b. To this relation, we associate an
undirected graph Γ(A) (which we call an orthograf), where the vertices are the
nonzero elements of the C∗-algebra A, with the identification of an element and
its scalar multiple; while there is an edge between two vertices a and b if a ⊥⊥S b.
We will show that for any C∗-algebra A, different from three simple
C∗-algebras, and for any two non-isolated vertices a and b in the orthograph,
we can find vertices c1, c2, c3 ∈ Γ(A) such that
a ⊥⊥S c1 ⊥⊥S c2 ⊥⊥S c3 ⊥⊥S b.
We will also describe the isolated vertices of the graph Γ(A) for any C∗-algebra
A. Finally, in the case of finite-dimensional C -algebras, we will determine the
diameter of Γ(A), i.e., the minimum number of elements required to connect any
two vertices. |