# Inner product space over a Hilbert C*-module

1. Mar 10, 2013

### CornMuffin

I am working on this problem, and I am having difficulties with a certain part of a proof:

If $A$ is a $C^*$-algebra. And $X$ is a Hilbert $A$-module. Can we say that $\langle X,X \rangle$ has an approximate identity $e_\alpha = \langle u_\alpha , v_\alpha \rangle$ such that $u_\alpha , v_\alpha$ are norm bounded by 1. Where $$\langle \cdot , \cdot \rangle$$ denotes the inner product.

I know that any $C^*$-algebra has a bounded approximate identity, and $\langle X,X \rangle$ is a $C^*$-algebra. So we can have $\| e_\alpha \| = \| \langle u_\alpha , v_\alpha \rangle \| \leq 1$