Invertibility of the operators on Hilbert spaces and ideals in \(C^*\)-algebras (Q2090532)

In the present paper, the author investigate invertibility of the operators on Hilbert spaces and Ideals in \(C^{\ast }\)-Algebras. Among interesting results, it was obtained a sufficient conditions for the positivity and invertibility of operators from \(\mathcal{B(H)},\) the \(\ast \)-algebra of all linear bounded operators on the Hilbert space \(\mathcal{H}.\) Also, for a von Neumann algebra \(\mathcal{A}\) it is proved that every arbitrary symmetry \(S\in \mathcal{A}\) is written as the product \(A^{-1}UA\) with a positive invertible \(A\) and a self-adjoint unitary \(U\) from \(\mathcal{A}\).