Public vivavoce Notification Date: Friday, 12 January 2024 Time: 11:00 AM Venue: Seminar Hall On the structure of certain noncommutative $C^{\ast}$algebras Malay Mandal Chennai Mathematical Institute. 120124 Abstract One of the most intriguing $C^{\ast}$algebra is the noncommutative unitary algebra $U^{nc}_n$, which is defined as a universal $C^{\ast}$algebra $A$ generated by $n^2$ elements that generate a unitary matrix in $M_n(A)$. In this thesis, we study three aspects of this $C^{\ast}$algebra $U^{nc}_n$. In the first part, we study several structural properties of this $C^{\ast}$algebra. In particular, we show that it possesses the Lifting property and is primitive. Also, we discuss the $RFD$ property of this $C^{\ast}$algebra. In the second part, we study the compact quantum semigroup structure of this $C^{\ast}$algebra $U^{nc}_n$ and the resulting compact semigroup structure of its state space, characterize all of its invertible elements. In the third part, we study the $KK$theory of this $C^{\ast}$algebra. In particular, we discuss the $KK$equivalnce of the $C^{\ast}$algebras $U^{nc}_n$ and $U^{nc}_{n,red}$, the reduced $C^{\ast}$algebra with respect to its natural state. In the last part, we study the simplicity property of mixed $q$Gaussian $C^{\ast}$algebra and the mixed $q$deformed ArakiWoods $C^{\ast}$algebra. More precisely, we discuss the simplicity of the mixed $q$deformed ArakiWoods $C^{\ast}$algebra $\Gamma_{T}(\mathcal{H}_{\mathbb{R}}, U_t)$ for a bounded infinitesimal generator $A$ of the oneparameter group of orthogonal representation $(U_t)_{t\in\mathbb{R}}$ and $U_t=A^{it},\ t\in\mathbb{R}$. All are invited to attend the vivavoce examination.
