Theorems · Definition · functional analysis

PositiveLinearMap.leftMulMapPreGNS

{A : Type u_1} →
  [inst : NonUnitalCStarAlgebra A] →
    [inst_1 : PartialOrder A] → (f : A →ₚ[ℂ] ℂ) → [inst_2 : StarOrderedRing A] → A → f.PreGNS →L[ℂ] f.PreGNS

The continuous linear map from a C⋆-algebra A to the PositiveLinearMap.preGNS space induced by a positive linear functional f : A →ₚ[ℂ] ℂ. This map is given by left-multiplication by a: x ↦ f.toPreGNS (a * f.ofPreGNS x). This is the map that is lifted to the completion of f.PreGNS (i.e. f.GNS) in order to define gnsNonUnitalStarAlgHom.

Defined in

Mathlib.Analysis.CStarAlgebra.GelfandNaimarkSegal

Cited by

4 results in Mathlib

Foundations

Depth 320 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NonUnitalCStarAlgebraPartialOrderStarOrderedRing