Estimates for the Involution of Decomposable Elements of a Complex Banach Algebra

An element $a$ of a complex Banach algebra with unit ${1\mspace{-4.85mu}{\mathrm I}}$ and with standard conditions on the norm ($\|ab\|\le\|a\|\cdot\|b\|$ and $\|{1\mspace{-4.85mu}{\mathrm I}}\|=1$) is said to be Hermitian if $\|e^{ita}\|=1$ for any real number $t$. An element is said to be decomposable if it admits a representation of the form $a+ib$ in which $a$ and $b$ are Hermitian. The decomposable elements form a Banach Lie algebra (with respect to the commutator). The Hermitian components are determined uniquely, and hence this Lie algebra has the natural involution $a+ib=x\to x^{*}=a-ib$. One can readily see that $\|x^{*}\|\le2\|x\|$. Among other things, we prove that $\|x^{*}\|\le\gamma \|x\|$, where $\gamma <2$. In fact, the situation is treated in more detail: the original problem is included in a continuous family parametrized by the numerical radius of the element. Finding the exact value of the constant $\gamma$ is reduced to a variational problem in the theory of entire functions of exponential type. Approximately, $\gamma$ is equal to $1.92\pm 0.04$.