Abstract
We consider a two parameter family of operators on a Hilbert space given by a multiplicative family of partial isometries with a generating subspace that commutes with an unitary group. The first parameter runs on a generalized interval of an abelian ordered group and the second on an abelian group. We show that it can be extended to a two parameter group of unitary operators on a larger Hilbert space. Applications to the problem of extending generalized semigroups of isometries and locally defined positive definite functions are given.
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