Estimating norms inC*-algebras of discrete groups

LetG be a discrete group, letK be a finite subset ofG and let χ K be the characteristic function ofK. Then χ K acts by convolution as a bounded operator onL2(G). We will prove that the norm |||χ K ||| of this operator always satisfies the following estimate: $$|||\chi _{\rm K} |||^2 \leqq k + 2\sqrt {w\left( {k - 1} \right)\left( {k - w} \right)} + \left( {k - 2} \right)\left( {k - w} \right)$$ . Here .