Is there any descent description of Morita theory for ultrametric Banach algebras?

To make this question more precise let $K$ be some completion of the field $\mathbb{Q}_p$ (I'm mostly interested in the norm completion of the algebraic completion $\mathbb{C}_p$ of $\mathbb{Q}_p$, and spherical completion $\Omega_p$ of $\mathbb{C}_p$). Recall that an (ultrametric) Banach space is $K$-linear space, complete with respect to an ultrametric norm on it, and ultrametric Banach algebras and modules are defined in the same spirit. For me it is important to have non-commutative Banach algebras.

Now, given two Banach algebras $A$ and $B$, it will be customary to call them Morita-equivalent if there is an isomorphism of categories of Banach modules over these algebras. My questions are:

  1. What could be a reasonable category of Banach modules over a Banach algebra, or, rather, what kind of morphisms should at best be taken. Does passing to bounded morphsisms instead of norm-decreasing change the picture that drastically?

  2. Evidently, a Morita-equivalence between such categories is to be embodied by a pair of bimodules ASB and BTA, such that the functors $-\hat\otimes_A S$ and $-\hat\otimes_B T$ implement an isomorphism of the categories of Banach modules. Here $\hat\otimes$ stands for projective tensor product. Recalling Rieffel's definition of strong Morita equivalence for $C^\ast$ algebras, I expect that some restrictions are to be taken into consideration in this definition, so I'd wish to know, what can they be.

Unfortunately, I can't find any more or less descriptive approach to such Morita equivalence in the literature. The things I've found mostly consider commutative Banach algebras over spherically complete fields. Has there been any sufficient progress since then?


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