Mixing
Direct limit of $L^2$ spaces
$begingroup$
Let $G$ be a discrete group and consider all the subgroups $N$ of finite index, these form a directed as follows: if $N subseteq N'$ (in this case we say that $N' < N$) there is a morphism $i_{N',N}:G/N to G/N'$ given by projection. The projective limit $widehat{G} = varprojlim_N G/N$ exists and it is called the profinite completion of $G$. On each quotient $G/N$ there is a normalized Haar measure $mu_N$ and the same is true for $widehat G$ with a measure $mu$. It is true that
$$L^2(widehat{G},mu) cong varinjlim_N L^2(G/N,mu_N)$$
where the morphism ${i_{N',N}}^*:L^2(G/N') to L^2(G/N)$ is given by composition with $i_{N',N}$?
measure-theory profinite-groups
asked Jan 2 at 18:39
k76u4vkweek547v7k76u4vkweek547v7
463316
$endgroup$
add a comment |
$begingroup$
Let $G$ be a discrete group and consider all the subgroups $N$ of finite index, these form a directed as follows: if $N subseteq N'$ (in this case we say that $N' < N$) there is a morphism $i_{N',N}:G/N to G/N'$ given by projection. The projective limit $widehat{G} = varprojlim_N G/N$ exists and it is called the profinite completion of $G$. On each quotient $G/N$ there is a normalized Haar measure $mu_N$ and the same is true for $widehat G$ with a measure $mu$. It is true that
$$L^2(widehat{G},mu) cong varinjlim_N L^2(G/N,mu_N)$$
where the morphism ${i_{N',N}}^*:L^2(G/N') to L^2(G/N)$ is given by composition with $i_{N',N}$?
measure-theory profinite-groups
asked Jan 2 at 18:39
k76u4vkweek547v7k76u4vkweek547v7
463316
$endgroup$
add a comment |
$begingroup$
Let $G$ be a discrete group and consider all the subgroups $N$ of finite index, these form a directed as follows: if $N subseteq N'$ (in this case we say that $N' < N$) there is a morphism $i_{N',N}:G/N to G/N'$ given by projection. The projective limit $widehat{G} = varprojlim_N G/N$ exists and it is called the profinite completion of $G$. On each quotient $G/N$ there is a normalized Haar measure $mu_N$ and the same is true for $widehat G$ with a measure $mu$. It is true that
$$L^2(widehat{G},mu) cong varinjlim_N L^2(G/N,mu_N)$$
where the morphism ${i_{N',N}}^*:L^2(G/N') to L^2(G/N)$ is given by composition with $i_{N',N}$?
measure-theory profinite-groups
asked Jan 2 at 18:39
k76u4vkweek547v7k76u4vkweek547v7
463316
$endgroup$
Let $G$ be a discrete group and consider all the subgroups $N$ of finite index, these form a directed as follows: if $N subseteq N'$ (in this case we say that $N' < N$) there is a morphism $i_{N',N}:G/N to G/N'$ given by projection. The projective limit $widehat{G} = varprojlim_N G/N$ exists and it is called the profinite completion of $G$. On each quotient $G/N$ there is a normalized Haar measure $mu_N$ and the same is true for $widehat G$ with a measure $mu$. It is true that
$$L^2(widehat{G},mu) cong varinjlim_N L^2(G/N,mu_N)$$
where the morphism ${i_{N',N}}^*:L^2(G/N') to L^2(G/N)$ is given by composition with $i_{N',N}$?
measure-theory profinite-groups
measure-theory profinite-groups
asked Jan 2 at 18:39
k76u4vkweek547v7k76u4vkweek547v7
463316
asked Jan 2 at 18:39
k76u4vkweek547v7k76u4vkweek547v7
463316
asked Jan 2 at 18:39
k76u4vkweek547v7k76u4vkweek547v7
463316
asked Jan 2 at 18:39
k76u4vkweek547v7k76u4vkweek547v7
463316
asked Jan 2 at 18:39
k76u4vkweek547v7k76u4vkweek547v7
463316
463316
add a comment |
add a comment |
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