Blecher, David P.2019-09-142019-09-14May 20192019-05May 2019Portions 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).https://hdl.handle.net/10657/4656An 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.application/pdfengThe author of this work is the copyright owner. UH Libraries and the Texas Digital Library have their permission to store and provide access to this work. UH Libraries has secured permission to reproduce any and all previously published materials contained in the work. Further transmission, reproduction, or presentation of this work is prohibited except with permission of the author(s).Jordan algebrasAlgebraOperator algebrasNoncommutative topologyOpen projection2019-09-14Thesisborn digital