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?

up vote 8 down vote accepted

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 non-unital 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

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