CONTINUOUS COALGEBRA ENDOMORPHISMS OFSOME COMPLETE ULTRAMETRIC HOPF ALGEBRAS

摘要

Due to properties of the topological tensor product of ultrametric Banachspaces, the algebraic notion of coalgebra has a natural ultrametric counter-part. An important class of ultrametric Banach coalgebras is provided by thespaces C(G, K) of continuous functions from a totally discontinuous compactgroup G with values in a complete ultrametric valued field K. The coproducton C(G, K) is induced by the law of multiplication of the group G. Related to ombral calculus is the characterization of the endomorphismsof a coalgebra. For a wide class of ultrametric Banach colagebras the monoidof the continuous coalgebra endomorphisms is anti-isomorphic to the monoidof continuous and weakly continuous algebra endomorphims of dual algebra.Hence, we recover earlier results obtained on the coalgebra C(Z, K), where Zpis the additive group of the ring of p-adic integers and K is a complete valuedfield of the field extension of p-adic numbers Qp. In addition for such groundfields K and more generally for K of residue characteristic p, we consider thecoalgebra C(Vq, K), where Vq is the infinite compact monothetic subgroup ofthe group of units of K generated by q which is not a root of unit such that — < 1, for h an integer > 1. One obtains, as for formal power series, aprocedure of substitution on the algebra of p-adic bounded measures M(Vq, K)which gives the continuous and weak*-continuous algebra endomorphisms ofM(Vq, K) and which in turn gives the continuous coalgebra endomorphisms ofC(Vq, K).