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Research articles

ScienceAsia 41(2015): 273-279 |doi: 10.2306/scienceasia1513-1874.2015.41.273

Operator connections and Borel measures on the unit interval

Pattrawut Chansangiama, Wicharn Lewkeeratiyutkulb,*

ABSTRACT:     A connection is a binary operation for positive operators satisfying monotonicity, the transformer inequality, and joint-continuity from above. A normalized connection is called a mean. Here we show that there is a one-to-one correspondence between connections and finite Borel measures on the unit interval via the integral representation in terms of weighted harmonic means with respect to that measure. This correspondence is affine and order-preserving. Hence every mean can be regarded as an average of weighted harmonic means. We also investigate decompositions of connections, means, symmetric connections and symmetric means.

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a Department of Mathematics, Faculty of Science, King Mongkut's Institute of Technology Ladkrabang, Bangkok 10520 Thailand
b Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330 Thailand

* Corresponding author, E-mail: wicharn.l@chula.ac.th

Received 21 Jul 2014, Accepted 20 May 2015