Exercise 32, § 6

Let G be a group. We say that G has the Powers property if, for every $n\in \mathbf{N}$ and every finite subset F of G$-\{e\}$, there exists a partition of G into two subsets A and B and a family $(g_i)_{1\leqslant i\leqslant n}$ of elements of G such that

(i) for every $g\in F$, the sets A and $gA$ are disjoint ;

(ii) for all distinct integers $j$ and $k$ between 1 and $n$, the sets $g_jB$ and $g_kB$ are disjoint.

Let G be a group possessing the Powers property. Let I be a nonzero closed two-sided ideal of the reduced stellar algebra Stell$_r(G)$ (exercise 25). Let $\pi :\mathbf{C}[G]\rightarrow$ Stell$_r(G)$ be the canonical representation.

a) There exists a family $(\lambda_g)_{g\in G-\{e\}}$ in $\mathbf{C}$ such that

is a hermitian element of Stell$_r(G)$ such that $y= 1 +x$ belongs to I. b) Let $\varepsilon >0$ be such that $\varepsilon <1$. There exists a finite subset F of G$-\{e\}$ and an integer $n\geqslant 1$ such that the element

$\surd$

satisfies $\|x'-x\|< \varepsilon$ and $\|x'\|<_2^n(1-\varepsilon )$.

c) Let $G = A\cup B$ and $(g_i)_{1\leqslant i\leqslant n}$ satisfying the conditions of the Powers property for the finite subset F and the integer $n$. Put

$'$ 1 $n'-1$

$y$ = $\sum\pi (g_i)x\pi (g_i)$.

$i=1$

We have $\|y'\|<1-\varepsilon$. (Let $p_i$ be the orthogonal projection of $\ell^2(G)$ onto $\ell^2(g_iB)$ ; verify that

for $1\leqslant i\leqslant n$, and deduce that

$i=1i=1$

then note that the endomorphisms $p_i\pi (g_i)x'\pi (g_i^{-1})$ have pairwise orthogonal images in order to estimate their norms.)

d) The ideal I contains the element

1 $^n-_1$

1 + $\sum\pi (g_i)x\pi (g_i)$,

$i=1$

and this element is invertible. Consequently, the stellar algebra Stell$_r(G)$ contains no nonzero closed two-sided ideal distinct from $\{0\}$ and Stell$_r(G)$.

e) Let G be a noncommutative free group. Then G has the Powers property.