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Orthogonality in $$C^{*}$$-algebras. (English) Zbl 1361.46017
Let $$A$$ be a $$C^*$$-algebra. For $$a, b \in A$$, we say that $$a$$ is algebraically orthogonal to $$b$$ if $$ab = 0 = ba = a^*b = ab^*$$. The author characterizes algebraic orthogonality between positive elements in a $$C^*$$-algebra in terms of positive linear functionals. He then applies this characterization to propose a generalized notion of orthogonality between positive elements with respect to certain subspaces of a $$C^*$$-algebra. He indeed describes the algebraic orthogonality in the context of $$\infty$$-orthogonality; see [A. K. Karn, Positivity 18, No. 2, 223–234 (2014; Zbl 1314.46026)]. He gives a norm and order theoretic characterization of algebraic orthogonality in some classes of $$C^*$$-algebras. He finally uses this generalization to propose a new kind of ordered spaces which characterize commutative $$C^*$$-algebras and von Neumann algebras. The paper may be regarded as a sequel to [A. K. Karn, Positivity 14, No. 3, 441–458 (2010; Zbl 1225.46014); ibid. 18, No. 2, 223–234 (2014; Zbl 1314.46026)].

##### MSC:
 46B40 Ordered normed spaces 46L05 General theory of $$C^*$$-algebras 46L30 States of selfadjoint operator algebras
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##### References:
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