Let $H$ be a Hilbert space. A subvector space $W\subset B(H)$ is called a fredholm subspace if there is an upper bound for the absolute value of Fredholm index of all Fredholm operatores $T$ in $W$.

Is there a classification of all $C^*$ algebras $A$ which admit an irreducible representation $\phi:A \to B(H)$ in some Hilbert space $H$ such that $\phi(A)$ is a Fredholm subspace of $B(H)$?

Is there a classification of all $C^*$ algebras $A$ which admit a faithful representation $\phi:A \to B(H)$ in some Hilbert space $H$ such that $\phi(A)$ is a Fredholm subspace of $B(H)$?

One can consider the terminology "Fredholm algebra" for any such $C^*$ algebras.