Maps on positive definite cones of 𝐶*-algebras preserving the Wasserstein mean
Lajos Molnár
Abstract
The primary aim of this paper is to present the complete description of the isomorphisms between positive definite cones of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">C^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-algebras with respect to the recently introduced Wasserstein mean and to show the nonexistence of nonconstant such morphisms into the positive reals in the case of von Neumann algebras without type I<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Subscript 2"> <mml:semantics> <mml:msub> <mml:mi/> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, I<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Subscript 1"> <mml:semantics> <mml:msub> <mml:mi/> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> direct summands. A comment on the algebraic properties of the Wasserstein mean relating associativity is also made.