Positive and conditionally positive linear functionals on coalgebras

  • Michael Schürmann
Conference paper
First Online:
Part of the Lecture Notes in Mathematics book series (LNM, volume 1136)

Abstract

For a linear functional ϕ on a complex coalgebra V the convolution exponential exp * ϕ can be defined. Let C⊂V be a set. Under which assumptions on C and ϕ is exp * (t ϕ) (v) positive for all v ∈ C and t≧0? We state a theorem for finite-dimensional coalgebras and apply it to two special cases. First we treat the case when V is the bialgebra of a semigroup G, and C is the convex cone in V of all positive functions on G. Then we formulate a theorem on sesquilinear forms on arbitrary complex coalgebras. In both cases exp * (t ϕ) is positive for all t≧0 if and only if ϕ is conditionally positive and hermitian. The theorem on sesquilinear forms on coalgebras is applied to linear functionals on certain graded bialgebras with an involution which we call skew graded *-bialgebras. The theory developed in this paper covers several known theorems and has applications to quantum probability.

Keywords

Convex Cone Closed Convex Cone Positive Definite Matrice Grade Vector Space Convolution Semigroup 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • Michael Schürmann
    • 1
  1. 1.Institut f. Ang. MathematikUniversität HeidelbergHeidelberg

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