# Fundamental theorem of coalgebras

Hi, Is there an "equivalent" to the fundamental theorem of coalgebras (any element of a coalgebra is contained in a finite dimensional sub-coalgebra) in the theory of algebraic quantum groups developed by Van Daele ? Thanks.

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Can you give a bit more background? What do you mean by "algebraic quantum groups"? What prevents the proof of that theorem from going through naively? – MTS Mar 8 '12 at 18:04
van Daele uses multipliers-- so $A$ is a non-unital algebra, and we have a non-degenerate homomorphism $\Delta:A\rightarrow M(A\otimes A)$ from $A$ into the multiplier algebra of $A\otimes A$. Then you have to think what coassociative should mean, and what the counit should do. (Which are pretty obvious). Often you want some sort of "regularity" condition. It seems to me hard to generalise from the usual proof (of the stated theorem), as now you have multipliers everywhere. But the question seems interesting, if a touch vague. – Matthew Daws Mar 8 '12 at 20:43
Ok thanks for these comments. I have read several proofs of the theorem in the "classical case" of co-algebras. Matthew Daws, you point out what prevent me from adapting naively these proofs. – Fanf Mar 12 '12 at 8:11