C Star Algebras

From MGSA

What are Fredholm operators?
What do they have to do with $K$ -theory for operator algebras?
Could you give some examples of interesting $C *$ -algebras with nontrivial $K$ -theory?
How does one recognize a compact operator? Give examples.
Prove that the Hilbert-Schmidt integral operators are compact.
One usually calls a $C *$ -algebra separable if it is represented on a separable Hilbert space. What are the $C *$ -algebras that are in fact separable as topological spaces?
State Kaplansky's Density Theorem. (Jones)
What is it good for? (e.g. in $L^\infty(S^1)$ ) (Jones)
Are the von Neumann algebras $l^\infty(\Z)$ and $l^\infty(S^1)$ isomorphic? Can they be embedded in a $I I 1$ factor? (Jones)
Define the index of a subfactor. (Jones)
What are all the hyperfinite subfactors of index $< 4$ ? (Jones)
Let $S$ be the unilateral shift. What is the commutant of $C * (S 2)$ ?
Do the Hilbert-Schmidt and trace class operators constitute $C *$ algebras under the Hilbert-Schmidt and trace norms, respectively?

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