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For any unital algebra $A$, we have an associated dual coalgebra $A^{\circ}$. (Recall that it is defined to be the largest subalgebra of the $\mathbf{C}$-linear dual of $A$ such that the coproduct $\Delta(f)(a,b) = f(ab)$ is well-defined.) What is the corresponding construction for a non-unital algebra. The coproduct part still works, but we have no counit, since this should arise as the dual of the unit. So are non-counital coalgebras studied in the literature? If so what are some references?

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The finite dual of a non-unital algebra has been introduced in Semiperfect and coreflexive coalgebras, S. Dăscălescu, M. C. Iovanov, Forum Math. 27 (2015), No. 5, 2587--2608. See also: arXiv:1512.09344 [math.RT].

For the (more general) notion of noncounital corings (and some relevant examples) you can also see ch.4, sect. 26 of Corings and comodules, T. Brzezinski, R. Wisbauer

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