We know that the norm of a unitary in a unital $C^*$algebra is one. Also, in a unital Banach algebra A, $u \in A$ is defined to be a unitary if $\u\ = \u^{1}\ =1$. I tried to prove it for the unitaries in unital Banach $*$algebra, but not able to get an answer. Is it true that a unitary in a unital Banach$*$algebra has norm one? Is there some counter example?
The answer is no even for a particular unitisation of the algebra of compact operators. We consider the Banach algebra $\widetilde{K}:= \{\lambda Id + k: k\text{ is compact}\}$, given with natural involution and norm $\\lambda Id + k\ := \lambda + \k\_{op}$ (it's not equal to the operator norm!); it is a unital Banach $\ast$algebra. The element $\left[\begin{array}{cccc} 0 & 1 & 0 & \cdots \\ 1 & 0 & 0 & \cdots \\ 0 & 0 & 1 & \cdots \\ \vdots & \vdots & \vdots &\ddots \end{array}\right]$ is unitary, but it's norm is equal to $3$.
If your question is:
suppose that $u^* u=1$. Must $u$ have norm one?
then no, as already the unit may be counterexample. Indeed, there exist nonunital semigroups $S$ whose semigroup algebras $\ell_1(S)$ are unital but the the unit has norm greater than one.

Thanks for the reply, but here we assume (in unital Banach $*$algebras) that unit has norm 1 – Ranjana Jain Sep 14 at 5:58