Geodesic
Estimates for the Involution of Decomposable Elements of a Complex Banach Algebra
Funkcionalʹnyj analiz i ego priloženiâ, Tome 39 (2005) no. 4, pp. 14-31 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $1.92\pm 0.04$.
Keywords: complex Banach algebra, involution, entire function, variational problem.
Mots-clés : decomposable element 
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E. A. Gorin. Estimates for the Involution of Decomposable Elements of a Complex Banach Algebra. Funkcionalʹnyj analiz i ego priloženiâ, Tome 39 (2005) no. 4, pp. 14-31. http://geodesic.mathdoc.fr/item/FAA_2005_39_4_a1/

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