# Lajos Molnar

## An Operation On The Positive Definite Cone of a $C^*$-Algebra and Its Algebraic Properties

Date Friday, June 6, 2014

The operation $Acirc B=sqrt{A}Bsqrt{A}$ defined for positive definite matrices or Hilbert space operators $A,B$ appears in different contexts and has applications for example in physics. We investigate the algebraic properties of $circ$ in the setting of $C^*$-algebras. We show that in that case the commutativity, associativity and distributivity of that operation are all equivalent. Furthermore, we present abstract characterizations for $circ$. If time permits we also show a few new characterizations of the commutativity of $C^*$-algebras in terms of particular algebraic properties of power functions, the logarithmic and exponential functions, and the sine and cosine functions.