Characterization of weak convergence in Hilbert $C^*$-modules?

Question

Assume $M$ is a Hilbert $C^*$-module and $(x_n)^{\infty }_{n=1}$ a bounded sequence in $M$. Are these equivalence?

1. $\langle x_n,y\rangle \to 0 $, for all $y\in M$.
2. $(x_n)$ is convergent to $0$ in the weak topology.

I know that these are equivalence in Hilbert spaces. But what about in Hilbert modules?

Answer 1

This is not true. Let $A$ be an infinite-dimensional unital $C^*$-algebra, and let $M=A$ with the inner product $\langle a,b\rangle=a^*b$. Then $\langle x_n,y\rangle \to 0$ for all $y\in A$ is equivalent to $(x_n)$ converging to $0$ in norm.