Theory of Jordan Operator Algebras and Operator *-Algebras



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An operator algebra is a closed subalgebra of B(H), for a complex Hilbert space H. By a Jordan operator algebra, we mean a norm-closed Jordan subalgebra of B(H), namely a norm-closed subspace closed under Jordan product a ◦ b = (ab + ba)/2. By an operator *-algebra we mean an operator algebra with a completely isometrically anti-isomorphic involution † making it a *-algebra. In this dissertation, we investigate the general theory of Jordan operator algebras and operator *-algebras. In Chapter 3, we present Jordan variants of ‘classical’ facts from the theory of operator algebras. For example we begin Chapter 3 with general facts about Jordan operator algebras. We then give an abstract characterizations of Jordan operator algebras. We also discuss unitization and real positivity in Jordan operator algebras. In Chapter 4, we study the hereditary subalgebras, open projections, ideals and M-ideals. We then develop the theory of real positive elements and real positive maps in the setting of Jordan operator algebras. Chapter 5 is largely concerned with general theory of operator *-spaces. In Chapter 6, we give several general results about operator -algebras. For example we prove some facts about involutions on nonselfadjoint operator algebras and their relationship to the C-algebra they generate. We also discuss contractive approximate identities, Cohen factorizations for operator *-algebras etc. In Chapter 7, we investigate hereditary subalgebras and ideals, noncommutative topology and peak projections in an involutive setting.



Jordan algebras, Algebra, Operator algebras, Noncommutative topology, Open projection


Portions of this document appear in: Blecher, David P., and Zhenhua Wang. "Involutive operator algebras." Positivity (2018): 1-41. And in: Blecher, David P., and Zhenhua Wang. "Jordan operator algebras: basic theory." Mathematische Nachrichten 291, no. 11-12 (2018): 1629-1654. And in: Blecher, David P., and Zhenhua Wang. "Jordan operator algebras revisited." arXiv preprint arXiv:1812.09995 (2018).