Thank you for reading.
My question was raised up when I tried to prove an example in the book of Liggett(1985), which is in P13 Example 2.3(a).
Here is a link of the page:
My questions are as follows:
1) what the definition of positive operator is while the book didn't mention that there is a Hilbert space;
2) how to prove Example 2.3 (a).
This question is not research level and would be better suited on MathStackExchange.
Positivity here just means $f\ge 0$ $\Rightarrow$ $Tf\ge 0$ (where $f\ge 0$ is defined as $f(x)\ge 0$ for each $x\in X$).
The example is verified using Proposition 2.2: If $f(\eta)=\min f(X)$ then $g=f-f(\eta)1\ge 0$ and since $T(1)=1$ we get $$ (T-I)(f)(\eta)=T(f)(\eta)-f(\eta)=T(f-f(\eta)1)(\eta)\ge 0.$$
