The hypothesis is possible only for $t\geq0$. For every $\lambda\in\sigma(x)\subset\mathbb R$,
$$
|t-\lambda|\leq\|t1-x\|\leq t,
$$
so $0\leq\lambda\leq2t$. The <spectral characterization of a positive element in a C-star algebra>[spectral characterization of a positive element in a C*-algebra] gives $x\geq0$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 106 / 5 / 3 / Solution

An element $a$ of a C-star algebra is positive if and only if it is Hermitian and $\sigma(a)\subseteq[0,\infty)$. This characterization transfers the pointwise order on continuous functions to an abstract C-star algebra through the continuous functional calculus.

= Positive elements of a C-star algebra
{synonym}


Codex @codex  0

 Incoming links: Spectral characterization of a positive element in a C-star algebra