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Order theory and interpolation in operator algebras. (English) Zbl 1357.47086

Let \(A\) be a norm-closed subalgebra of \(B(H)\), where \(H\) is a Hilbert space. The authors continue their study of new notions of positivity for general operator algebras in the above setting, which was initiated in their earlier work [J. Funct. Anal. 261, No. 1, 188–217 (2011; Zbl 1235.47087); ibid. 264, No. 4,1049–1067 (2013; Zbl 1270.47067)] and in [J. Math. Anal. Appl. 381, No. 1, 202–214 (2011; Zbl 1235.47090)] by the second author.
The paper under review is a thorough and systematic study of the foundational aspects of positivity and the associated ordering for operator algebras, which is derived from the closed cone \(r_A = \{a \in A: a + a^* \geq 0\}\) of accretive operators (as well as certain subcones). Particular attention is paid to the order relations of \(A\) and the \(C^*\)-algebra it generates. The paper contains a wealth of material, including several applications to non-commutative topology, non-commutative peak sets, lifting problems for peak projections, peak interpolation and approximate identities. For instance, they obtain non-commutative versions for operator algebras of the strict Urysohn lemma and the Tietze extension theorem.

MSC:

47L30 Abstract operator algebras on Hilbert spaces
46B40 Ordered normed spaces
46L52 Noncommutative function spaces
46L85 Noncommutative topology
47L07 Convex sets and cones of operators
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
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