Public viva-voce Notification Date: Friday, 12 January 2024 Time: 11:00 AM Venue: Seminar Hall On the structure of certain non-commutative $C^{\ast}$-algebras Malay Mandal Chennai Mathematical Institute. 12-01-24 Abstract One of the most intriguing $C^{\ast}$-algebra is the non-commutative 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 Araki-Woods $C^{\ast}$-algebra. More precisely, we discuss the simplicity of the mixed $q$-deformed Araki-Woods $C^{\ast}$-algebra $\Gamma_{T}(\mathcal{H}_{\mathbb{R}}, U_t)$ for a bounded infinitesimal generator $A$ of the one-parameter 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 viva-voce examination.
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