$<$ and $>$ symbols used with matrices

Question

In this Wikipedia page, in the first property, it says $a_1$ < $a_2$, where $a_1$ and $a_2$ are elements of a C*-algebra.

The easiest way for me to think of $a_1$ and $a_2$ are as matrices. Does that statement mean that the smallest eigenvalue of $a_1$ is lesser than the smallest eigenvalue of $a_2$?

Answer 1

In terms of matrices, $a_1\leq a_2$ means that $a_2-a_1$ is positive semidefinite. The symbols $>$ and $<$ are rarely used, because their meaning is confusing: when you write $a>0$, you could mean that

• $a\geq0$ and $a\ne0$,

but it is common to reserve $a>0$ to denote that

• $a$ is positive definite.

The notion is only related to eigenvalues when $a_1,a_2$ are selfadjoint. In general, either you have a given C$^*$-algebra with its positive cone given or, as you can always do, you have your algebra represented on $B(H)$ and then $a\geq0$ means $$ \langle a\xi,\xi\rangle\geq0\ \ \ \text{ for all } \xi\in H. $$

For matrices, the above is equivalent to $a=a^*$ and all the eigenvalues of $a$ non-negative.