Invertibility of the Operators on Hilbert Spaces and Ideals in $C^*$-Algebras

Let $\mathcal{H}$ be a Hilbert space over the field $\mathbb{C}$, and let $\mathcal{B}(\mathcal{H})$ be the $\ast$-algebra of all linear bounded operators in $\mathcal{H}$. Sufficient conditions for the positivity and invertibility of operators from $\mathcal{B}(\mathcal{H})$ are found. An arbitrary symmetry from a von Neumann algebra $\mathcal{A}$ is written as the product $A^{-1}UA$ with a positive invertible $A$ and a self-adjoint unitary $U$ from $\mathcal{A}$. Let $\varphi$ be the weight on a von Neumann algebra $\mathcal{A}$, let $A\in \mathcal{A}$, and let $\|A\|\le 1$. If $A^*A-I\in \mathfrak{N}_{\varphi}$, then $|A|-I\in \mathfrak{N}_{\varphi}$ and, for any isometry $U\in \mathcal{A}$, the inequality $\|A-U\|_{\varphi,2}\ge \||A|-I\|_{\varphi,2}$ holds. If $U$ is a unitary operator from the polar decomposition of the invertible operator $A$, then this inequality becomes an equality.