zbMATH — the first resource for mathematics

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)].

46B40 Ordered normed spaces
46L05 General theory of \(C^*\)-algebras
46L30 States of selfadjoint operator algebras
Full Text: DOI
[1] Birkhoff, G, Orthogonality in linear metric spaces, Duke Math. J., 1, 169-172, (1935) · Zbl 0012.30604
[2] James, RC, Orthogonality in normed linear spaces, Duke Math. J., 12, 291-302, (1945) · Zbl 0060.26202
[3] Kadison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras, I. Academic Press, New York (1983) · Zbl 0518.46046
[4] Karn, AK, A \(p\)-theory of ordered normed spaces, Positivity, 14, 441-458, (2010) · Zbl 1225.46014
[5] Karn, AK, Orthogonality in \(ℓ _p\)-spaces and its bearing on ordered normed spaces, Positivity, 18, 223-234, (2014) · Zbl 1314.46026
[6] Oikhberg, T; Peralta, AM; Ramirez, M, Automatic continuity of \(M\)-norms on \(C^*\)-algebras, J. Math. Anal. Appl., 381, 799-811, (2011) · Zbl 1225.46043
[7] Oikhberg, T; Peralta, AM, Automatic continuity of orthogonality preservers on a non-commutative \(L_p (τ )\) space, J. Funct. Anal., 264, 1848-1872, (2013) · Zbl 1288.47034
[8] Pedersen, G.K.: \(C^{⁎ }\)-Algebras and Their Automorphism Groups. Academic Press, London (1979) · Zbl 0416.46043
[9] Raynaud, Y; Xu, Q, On subspaces of non-commutative \(L^p\)-spaces, J. Funct. Anal., 203, 149-196, (2003) · Zbl 1056.46056
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.