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.
