# Which Banach spaces have categorical duals?

I was looking carefully at all the definitions, trying to understand exactly what was going on in this question on categorical duals in Banach spaces. It seems that in the category of Banach spaces with contractions, only finite dimensional Banach spaces have duals in the sense of category theory. This is because one needs morphisms

$$I \to A \otimes A^*$$

and

$$A \otimes A^* \to I$$

where the first is like the inclusion of the identity and the second like the trace. This means that the identity map on $A$ has to have a trace, and that only works for finite dimensional Banach spaces.

So, my question: am I right? Or have I misunderstood something here?

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From the recent discussion in the question you linked to, it seems that even (some? most?) finite-dimensional Banach spaces do not have categorical duals in the precise sense you describe. In fact it's clear what goes wrong here with the construction used in the algebraic setting of finite-dimensional vector spaces: the composition of the two maps you describe would be multiplication on $I$ by the dimension of $A$, but that is not a contraction unless $\dim A \le 1$!