# Is $\ell_2(A)$ a Hilbert C$^*$-module with Opial property?

If $$A=Mat_{n\times n}(\mathbb{C})$$, Is $$\ell_2(A)$$ a Hilbert $$A$$-module with Opial property?

Opial property: If ($$w-\lim x_n=0$$) then $$(\liminf \lVert x_n\rVert<\liminf \lVert x_n-y \rVert$$) for all $$y\ne 0$$.

I tried to solve it for many hours, But I didn't solve it.

• Do you have a definition for this property ? – Bleuderk Nov 20 at 15:02
• The answer is yes. Hint, think about why ordinary $l^2$ has this property. The two cases are very similar. – Nik Weaver Nov 20 at 16:50
• And waht is the Opial property for $A$ Hilbert modules ? Do we consider the topology associated to the $A$ linear continuous maps $\phi : M \to A$ for the weak topology? – Bleuderk Nov 20 at 17:46
• The Opial property in the Hilbert A-module $\ell_2(A)$ is equivalent to $\liminf \lVert \langle x_n,x_n\rangle \rVert < \liminf \lVert \langle x_n,x_n\rangle +\langle y,y \rangle \rVert$ for all $y\ne 0$. – Dadrahm Nov 21 at 8:33