Schröder–Bernstein theorems for operator algebras

The Schröder–Bernstein theorem, traditionally framed in set theory, states that if there are injective (one-to-one) functions \( f: A \to B \) and \( g: B \to A \) between two sets \( A \) and \( B \), then there exists a bijection (one-to-one and onto function) between \( A \) and \( B \).