Determine if a sequence of projections converging in strong operator topology also converges in the weak *-topology | Step-by-Step Solution

Understanding Convergence in Different Topologies on C*-Algebras

What We're Solving: We need to determine whether a sequence of projections that converges in the strong operator topology (SOT) will also automatically converge in the weak *-topology.

The Approach: We'll approach this by:

• 1. Understanding what each topology means for convergence
• 2. Comparing their relative "strength"
• 3. Using the specific properties of projections and C*-algebras
• 4. Either proving the statement or finding a counterexample

The key insight is that we need to understand how these topologies relate to each other - which one has more open sets (is finer)?

Step-by-Step Solution:

Step 1: Recall the definitions

• Strong operator topology (SOT): A net $(p_α)$ of operators converges to $p$ if $\|p_α x - px\| \to 0$ for all $x$ in the Hilbert space
• Weak -topology: This is the weakest topology making all functionals of the form $a \mapsto \phi(a)$ continuous, where $\phi$ ranges over the dual space $A^$

Step 2: Compare the topologies The weak *-topology is generally weaker than the strong operator topology! This means:

• Weak *-topology has fewer open sets
• Any weak *-convergent sequence is SOT-convergent
• But SOT-convergent sequences might not be weak *-convergent

Step 3: Apply this to our projections Since SOT is stronger (finer) than the weak *-topology, we have: $$\text{SOT convergence} \Rightarrow \text{weak *-topology convergence}$$

This is a general principle: convergence in a stronger topology implies convergence in any weaker topology.

Step 4: Verify with projection properties Projections are self-adjoint idempotents ($p^2 = p = p^$). In a C-algebra, if projections $p_n \to p$ in SOT, then for any state $\phi \in A^*$: $$\phi(p_n) \to \phi(p)$$ This confirms weak *-convergence.

The Answer: Yes! If a sequence of projections converges in the strong operator topology, it automatically converges in the weak -topology. This follows from the fundamental fact that SOT is finer than the weak -topology.

Memory Tip: Remember "Strong implies Weak" - just like in real life! If you can lift something heavy (strong convergence), you can certainly handle something light (weak convergence). The strong operator topology has more "muscle" (more open sets) than the weak *-topology.

Great question! This really gets to the heart of how different topologies on operator algebras relate to each other. Understanding these relationships is crucial for working with von Neumann algebras and C*-algebras. Keep exploring these beautiful connections! 🌟