Definition 6

Let A be an involutive algebra. An element $x$ of A is said to be positive if it is hermitian and if Sp$'_A(x)\subset \mathbf{R}_+$. The set of positive elements of A is denoted by $A_+$. It is a subset of $A_h$.

One writes $x\geqslant y$ if $x-y\in A_+$.

If the involutive algebra A is unital, its unit element is positive.

If B is an involutive subalgebra of A, one has $B_+= B\cap A_+$ (cor. of prop. 5 of I, p. 106).

If $\pi : A\rightarrow$ B is a morphism of involutive algebras, then $\pi (A_+)\subset B_+$.