Browsing by Author "Banik, Sayan Kansa"

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On Universal C∗ Algebras associated to Operator Spaces and Generalized Crossed Products
(2026-07-08) Banik, Sayan Kansa
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.