On Universal C∗ Algebras associated to Operator Spaces and Generalized Crossed Products

Abstract

In my thesis, we study generalized crossed product constructions of the group \( C^* \)-algebra \( C^*(G) \) with respect to certain completely positive maps, where \( G \) is assumed to be a discrete amenable group. We also investigate the universal \( C^* \)-algebra \( \mathcal{E}_\alpha \) introduced by Hirshberg, which is constructed from \( C^*(G) \) and a pure injective homomorphism \( \alpha \colon G \to G \). In particular, we analyze its relationship with Exel's construction of generalized crossed products associated with the endomorphism of \( C^*(G) \) induced by \( \alpha \), together with an appropriate choice of transfer operator. In addition, we study crossed products of \( C^*(G) \) arising from states and conditional expectations of the form \( E_H \colon C^*(G) \to C^*(H) \), where \( H < G \) is a proper subgroup. We examine how generalized crossed product constructions change when passing from endomorphism-based framework to to the case of completely positive maps. We study crossed products of \( C^* \)-algebras with respect to states, focusing in particular on \( C^*(G) \) equipped with its canonical trace and construct a spatial representation of the system isomorphic to the universal crossed product construction. Further, we show that when \( G \) is virtually abelian, there exists no spatial representation of the system \( (C^*(G), E_H) \) inside \( B(\ell^2(G)) \). Finally, we construct a specific spatial representation of this system and show that, when $[G:H]=\infty$ , the resulting spatial \( C^* \)-algebra is isomorphic to the corresponding universal crossed product In the other half, we make a detailed study of operator spaces associated to Brown's noncommutative unitary $C^\ast$-algebra $\mathcal{U}^{nc}_n$ and related $C^\ast$-algebras. Specifically, we identify the universal $C^{\ast}$-algebra $C^{\ast}\langle M_n(\C)^{\ast}\rangle$ of the operator space $M_n(\C)^{\ast}$ with the non-commutative $C^{\ast}$-algebra $\Y^{nc}_n$, the universal unital $C^{\ast}$-algebra generated by elements $u_{ij}$, $1\leq i,j\leq n$ satisfying the relations which make $[u_{ij}]$ a contractive matrix. We also exhibited several operator algebraic properties of $C^{\ast}\langle M_n(\C)^{\ast}\rangle$- in particular, we study the Lifting property (LP), residual finite dimensionality and primitivity of $C^{\ast}\langle M_n(\C)^{\ast}\rangle$. Further, we study the maximal and minimal operator space structure of the standard generators of $\U^{nc}_n$ as well as $\U^{nc}_{n, red}$. Finally, we discuss a natural compact quantum semigroup structure on $C^{\ast}\langle M_n(\C)^{\ast}\rangle$, characterizing invertible elements in its state space under convolution.