Compact Group Actions on C*-algebras: Classification, Non-Classifiability and Crossed Products and Rigidity Results for Lp-operator Algebras
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This dissertation is concerned with representations of locally compact groups on different classes of Banach spaces. The first part of this work considers representations of compact groups by automorphisms of C*-algebras, also known as group actions on C*-algebras. The actions we study enjoy a freeness-type of property, namely finite Rokhlin dimension. We investigate the structure of their crossed products, mainly in relation to their classifiability, and compare the notion of finite Rokhlin dimension with other existing notions of noncommutative freeness. In the case of Rokhlin dimension zero, also known as the Rokhlin property, we prove a number of classification theorems for these actions. Also, in this case, much more can be said about the structure of the crossed products. In the last chapter of this part, we explore the extent to which actions with Rokhlin dimension one can be classified. Our results show that even for Z_2-actions on O_2, their classification is not Borel, and hence it is intractable. The second part of the present dissertation focuses on isometric representations of groups on Lp-spaces. For p=2, these are the unitary representations on Hilbert spaces. We study the Lp-analogs of the full and reduced group \ca s, particularly in connection to their rigidity. One of the main results of this work asserts that for p different from 2, the isometric isomorphism type of the reduced group Lp-operator algebra recovers the group. Our study of group algebras acting on Lp-spaces has also led us to answer a 20-year-old question of Le Merdy and Junge: for p different from 2, the class of Banach algebras that can be represented on an Lp-space is not closed under quotients. We moreover study representations of groupoids, which are a generalization of groups where multiplication is not always defined. The algebras associated to these objects provide new examples of Lp-operator algebras and recover some previously existing ones. Groupoid Lp-operator algebras are particularly tractable objects. For instance, while groupoid Lp-operator algebras can be classified by their K_0-group (an ordered, countable abelian group), we show that UHF-Lp-operator algebras not arising from groupoids cannot be classified by countable structures. This dissertation includes unpublished coauthored material.