I‘m currently reading Arveson’s “A Short Course on Spectral Theory”, and I’m stuck at Exercise 3.1 (1). The question is:

Let $l^{\infty}(\mathbb{N})$ be the set of all bounded sequences of complex numbers. A Banach limit is a linear functional $\Lambda : l^{\infty}(\mathbb{N}) \to \mathbb{C}$ which satisfies $|| \Lambda ||=\Lambda((1,1,\ldots))=1$, and $\Lambda (Ta) = \Lambda(a)$ for any $a\in l^{\infty}(\mathbb{N})$, where $T$ denotes the left shift operator.

Prove that every Banach limit $\Lambda$ is a positive linear functional in the sense that $$ \forall n \ a_n\geq 0 \implies \Lambda(\{a_n\}_n) \geq 0 $$

I have no idea to prove it. Would you please give me some hints?

I‘m currently reading Arveson’s “A Short Course on Spectral Theory”, and I’m stuck at Exercise 3.1 (1). The question is:

Let $l^{\infty}(\mathbb{N})$ be the set of all bounded sequences of complex numbers. A Banach limit is a linear functional $\Lambda : l^{\infty}(\mathbb{N}) \to \mathbb{C}$ which satisfies $|| \Lambda ||=\Lambda((1,1,\ldots))=1$, and $\Lambda (Ta) = \Lambda(a)$ for any $a\in l^{\infty}(\mathbb{N})$, where $T$ denotes the left shift operator.

Prove that every Banach limit $\Lambda$ is a positive linear functional in the sense that $$ \forall n \ a_n\geq 0 \implies \Lambda(\{a_n\}_n) \geq 0 $$

I have no idea to prove it. Would you please give me some hints?