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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.

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  • $\begingroup$ Do you have a definition for this property ? $\endgroup$
    – InfiniteLooper
    Commented Nov 20, 2018 at 15:02
  • $\begingroup$ en.wikipedia.org/wiki/Opial_property $\endgroup$
    – Jochen Wengenroth
    Commented Nov 20, 2018 at 15:03
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    $\begingroup$ The answer is yes. Hint, think about why ordinary $l^2$ has this property. The two cases are very similar. $\endgroup$
    – Nik Weaver
    Commented Nov 20, 2018 at 16:50
  • $\begingroup$ 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? $\endgroup$
    – InfiniteLooper
    Commented Nov 20, 2018 at 17:46
  • $\begingroup$ 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 $y\ne 0$. $\endgroup$
    – Darman
    Commented Nov 21, 2018 at 8:33

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