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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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