$C^*$ algebras and bounded operators on $H$

Question

If $A$ is any $C^*$ algebra,then it is isomorphic to a subalgebra of $B(H)$.My question is :If $A$ is an infinite dimensional algebra,does there must exist an infinite dimensional Hilbert space $ H_0$ such that $A$ is $*$ isomorphic to $B(H_0)$

Answer 1

If $A$ is infinite dimensional, it will necessarily embed as a subalgebra of some $B(H)$ for some infinite dimensional Hilbert space $H$, as $H$ finite dimensional implies $B(H)\cong \mathbb{M}_n(\mathbb{C})$, for some $n\in \mathbb{N}$, which would imply that $A$ is finite dimensional.

However, $A$ need not be the whole space $B(H)$. As Robert said, there are commutative $C^*$ algebras, and except in the one dimensional case, $B(H)$ is not commutative.

Answer 2

No. For example, there are commutative $C^*$ algebras.