Geometric Aspects of $C^*$-Extreme Points II

Building on the results of \cite{HR}, we investigate the relationship between $C^*$-extremality and linear extremality in the closed unit balls of general unital $C^*$-algebras. We show that, unlike in von Neumann algebras, these two notions of extremality need not coincide in general $C^*$-algebras. More precisely, we prove that the unilateral shift is a linear extreme point, but not a $C^*$-extreme point, of the closed unit ball of the Toeplitz $C^*$-algebra. We further establish that $C^*$-extremality implies linear extremality whenever the $C^*$-algebra admits a faithful unital representation on a separable Hilbert space. We also prove this implication for elements admitting a factorization $x=v|x|$, where $v$ is a partial isometry belonging to the algebra, and extend the result to suitable quotient $C^*$-algebras, including the Calkin algebra. Finally, we characterize the linear extreme points of the closed unit ball of unital $C^*$-subalgebras of $B(H)$ containing the compact operators.