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 $C^*$-algebra, $T$ be a locally compact Hausdorff space equipped with a Radon measure $μ$ 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 $t \mapsto \|A_t\|$ is integrable. If $α: T \times T \to \mathbb{C}$ is a measurable function such that $\bar{α(t,s)}α(s,t)=1$ for all $t, s \in T$, then we show that \begin{align*} \int_T\int_T&\left|α(t,s) A_t-α(s,t) A_s\right|^2dμ(t)dμ(s)+\int_T\int_T\left|α(t,s) B_t-α(s,t) B_s\right|^2dμ(t)dμ(s) \nonumber &= 2\int_T\int_T\left|α(t,s) A_t-α(s,t) B_s\right|^2dμ(t)dμ(s) - 2\left|\int_T(A_t-B_t)dμ(t)\right|^2\,. \end{align*}

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