On the variance of measurable operators

We study the block projection operator Pₙ , n ≥ 2 , in the $$*$$ -algebra S(M, τ) of all τ -measurable operators, where τ is a faithful normal semi-finite trace on the von Neumann algebra M . For a state τ , we study the variance D of an operator from L₂(M,τ) . These results are new, even for the $$*$$ -algebra M=B(H) of all bounded linear operators in H equipped with canonical trace τ =tr . For a random bounded linear operator A in a Hilbert space H , we consider the operator-valued random variance D(A) , which is used to investigate the validity of the law of large numbers for the product of independent identically distributed semigroups of bounded linear operators. We present conditions on a bounded self-adjoint operator X∈ B(H) and on a random operator A , whose variances are related as D(X)=τ (D(A)) .