Fejér monotonicity is a well-established property commonly observed in sequences generated by optimization algorithms. In this paper, we introduce an extension of this property, called Fejér* monotonicity, which was initially proposed in [SIAM J. Optim., 34(3), 2535-2556 (2024)]. We discuss and build upon the concept by exploring its behavior within Hilbert spaces, presenting an illustrative example and insightful results regarding weak and strong convergence. We also compare Fejér* monotonicity with other weak notions of Fejér-like monotonicity, to better establish the role of Fejér* monotonicity in optimization algorithms.