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

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 $$B(H_0)$$

## 2 Answers

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.

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