Abstract
We describe the class of translation invariant measures on the algebra ℬ(ℋ) of bounded linear operators on a Hilbert space ℋ and some of its subalgebras. In order to achieve this we apply two steps. First we show that a total minimal system of finite weights on the operator algebra defines a family of rectangles in this algebra through construction of operator intervals. The second step is construction of a translation invariant measure on some subalgebras of algebra ℬ(ℋ) by the family of rectangles. The operator intervals in the Jordan algebra ℬ(ℋ)sa is investigated. We also obtain some new operator inequalities.
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