‎strictly ‎positive elements

Question

Let ‎$‎‎A$ ‎be a ‎‎‎‎$‎‎C^*$-algebra‎.

‎$‎‎a\in A^+$ ‎is ‎strictly ‎positive in ‎$‎‎A$‎ ‎if ‎‎$‎‎‎\overline{aAa}=A‎$‎‎

*I know that if $A$ is unital, $a\in A^+$ is strictly positive iff $a\in Inv(A)$

Q1:Let ‎$‎‎A:=C_0(0,1)$‎. Is there any strictly positive element is‎$‎‎A$‎?why? ‎

Q2:Let $A$ be non-unital. Is there any condition (like *) for $a\in A^+$ such that $a$ is strictly positive ?

Answer 1

1. Let $a(t)>0$ for all $t\in (0,1)$ (eg $1-2|t-1/2|$). Then for each $n \in \mathbb{N}$, there exists a function $c_n \in C_0(0,1)$ so that $c_n(t)a(t)=1$ whenever $t \in I_n :=[\frac{1}{n},1-\frac{1}{n}]$. This consideration shows that for any $f \in C_0(0,1)$ $(a c_n f c_n a)(t)=f(t)$ whenever $t \in I_n$. The construction implies that $f \in \overline{aAa}$. (Where $\lim_{t\to 0/1}f(t)=0$ is necessary for the convergence of the sequence in sup-norm sense).

   Since any function that is always positive has hermitian root functions, whenever $a(t)>0$ for all $t$, then $a \in A^+$ and as such $a$ is actually strictly positive. In fact only if a function in $C_0(0,1)$ has strictly positive range is it strictly positive in the $C^*$ algebra sense.

2. In this script there are some equivalent notions of when elements are strictly positive. Specifically if $a \in A^+$ then $\overline{aAa}=A \iff \overline{aA}=A \iff \left(\frac{a}{||a||}\right)^{1/n}$ is an approximate identity.

Answer 2

Let $f\in A=C_0(0,1)$. Then

$f(t)>0$ for all $t$ if and only if $f$ is strictly positive.

Assume first that $f(t)>0$ for all $t$. Fix, initially, $g\in C_0(0,1)$ such that $g(t)=0$ for all $t\in (0,b)\cup(1-b,1)$ for some $b>0$. As $[b,1-b]$ is compact and $f>0$ on $[b,1-b]$, there exists $\delta>0$ with $f(t)\geq\delta$ for all $t\in [b,1-b]$. Let $$ h(t)=\begin{cases}0,&\ t\in(0,b),\\ \ \\\dfrac{g(t)}{f(t)^2},&\ t\in[b,1-b],\\ \ \\0,&\ t\in(1-b,1)\end{cases} $$ Then $h\in C_0(0,1)$ and $g=fhf\in fAf$. As the compact supported functions are dense in $A$, we get that $A=\overline{fAf}$, so $f$ is strictly positive.

Conversely, if $f(t_0)=0$ for some $t_0\in(0,1)$, then all functions in $\overline{fAf}$ vanish at $t_0$, and so $\overline{fAf}\subsetneq A$. This shows that $f$ is not strictly positive.

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As for your second questions, the only characterization I know is that $a\in A_+$ is strictly positive if and only if $\{(a/\|a\|)^{1/n}\}$ is an approximate unit for $A$.