An operator extension of the parallelogram law and related norm inequalities

We establish a general operator parallelogram law concerning a characterization of inner product spaces, get an operator extension of Bohr’s inequality and present several norm inequalities. More precisely, let $\mathfrak{A}$ be a $C^{\ast }$-algebra, $T$ be a locally compact Hausdorff space equipped with a Radon measure $\mu$ and let $(A_t{)}_{t\in T}$ be a continuous field of operators in $\mathfrak{A}$ such that the function $t\mapsto A_t$ is norm continuous on $T$ and the function $t\mapsto \mathrm{\parallel }{A}_t\mathrm{\parallel }$ is integrable. If $\alpha :T\times T\to \mathbb{C}$ is a measurable function such that $\overline{\alpha (t,s)}\alpha (s,t)=1$ for all $t,s\in T$, then we show that \begin{align*} \int_T\int_T&\left|\alpha(t,s) A_t-\alpha(s,t) A_s\right|^2d\mu(t)d\mu(s)+\int_T\int_T\left|\alpha(t,s) B_t-\alpha(s,t) B_s\right|^2d\mu(t)d\mu(s) \nonumber &= 2\int_T\int_T\left|\alpha(t,s) A_t-\alpha(s,t) B_s\right|^2d\mu(t)d\mu(s) - 2\left|\int_T(A_t-B_t)d\mu(t)\right|^2,. \end{align*}