Almost having invariant vectors vs having almost invariant vectors?
$begingroup$
Let $Gamma$ be a discrete and countable Group and let $pi:Gammato mathcal{B(H)}$ be a unitary representation.
We say that $pi$ almost has invariant vectors if for every compact (=finite) subset $FsubseteqGamma$ and $varepsilon >0$ there there exists a unit vector $xiinmathcal{H}$ such that
$$ |pi(s)xi -xi|<varepsilon; : forall sin F.$$
This is demonstrated to be equivalent to the weak containment of the trivial representation, denoted by $1_Gamma$, in $pi$ (see for example Cor. F.1.5, [1], general case of locally compact groups).
In other literature (for example Thm. A.5. and A.12, [2]) I have found the following definition:
$(pi,mathcal{H})$ admits almost invariant vectors if there exists a sequence of unit vectors $(xi_n)_nsubseteq mathcal{H}$ such that
$$|pi(s)xi_n-xi_n| to 0 ; : ; forall s in Gamma$$
which is also claimed to be equivalent to the trivial representation being weakly contained in $pi$.
Are these conditions really equivalent? If so, how would I go about understanding this equivalence? At this point I don't see how these two could be equivalent, without considering nets instead of sequences in the second condition.
Many thanks in advance.
[1] "Kazhdan’s Property (T)" by B. Bekka, P. de la Harpe and A. Valette)
[2] "Amenability of discrete groups by examples"
by Kate Juschenko
operator-theory representation-theory c-star-algebras
edited Jan 13 at 23:57
Bernard
124k742117
asked Jan 13 at 23:47
OpalgalOpalgal
595
$endgroup$
add a comment |
$begingroup$
Let $Gamma$ be a discrete and countable Group and let $pi:Gammato mathcal{B(H)}$ be a unitary representation.
We say that $pi$ almost has invariant vectors if for every compact (=finite) subset $FsubseteqGamma$ and $varepsilon >0$ there there exists a unit vector $xiinmathcal{H}$ such that
$$ |pi(s)xi -xi|<varepsilon; : forall sin F.$$
This is demonstrated to be equivalent to the weak containment of the trivial representation, denoted by $1_Gamma$, in $pi$ (see for example Cor. F.1.5, [1], general case of locally compact groups).
In other literature (for example Thm. A.5. and A.12, [2]) I have found the following definition:
$(pi,mathcal{H})$ admits almost invariant vectors if there exists a sequence of unit vectors $(xi_n)_nsubseteq mathcal{H}$ such that
$$|pi(s)xi_n-xi_n| to 0 ; : ; forall s in Gamma$$
which is also claimed to be equivalent to the trivial representation being weakly contained in $pi$.
Are these conditions really equivalent? If so, how would I go about understanding this equivalence? At this point I don't see how these two could be equivalent, without considering nets instead of sequences in the second condition.
Many thanks in advance.
[1] "Kazhdan’s Property (T)" by B. Bekka, P. de la Harpe and A. Valette)
[2] "Amenability of discrete groups by examples"
by Kate Juschenko
operator-theory representation-theory c-star-algebras
edited Jan 13 at 23:57
Bernard
124k742117
asked Jan 13 at 23:47
OpalgalOpalgal
595
$endgroup$
add a comment |
$begingroup$
Let $Gamma$ be a discrete and countable Group and let $pi:Gammato mathcal{B(H)}$ be a unitary representation.
We say that $pi$ almost has invariant vectors if for every compact (=finite) subset $FsubseteqGamma$ and $varepsilon >0$ there there exists a unit vector $xiinmathcal{H}$ such that
$$ |pi(s)xi -xi|<varepsilon; : forall sin F.$$
This is demonstrated to be equivalent to the weak containment of the trivial representation, denoted by $1_Gamma$, in $pi$ (see for example Cor. F.1.5, [1], general case of locally compact groups).
In other literature (for example Thm. A.5. and A.12, [2]) I have found the following definition:
$(pi,mathcal{H})$ admits almost invariant vectors if there exists a sequence of unit vectors $(xi_n)_nsubseteq mathcal{H}$ such that
$$|pi(s)xi_n-xi_n| to 0 ; : ; forall s in Gamma$$
which is also claimed to be equivalent to the trivial representation being weakly contained in $pi$.
Are these conditions really equivalent? If so, how would I go about understanding this equivalence? At this point I don't see how these two could be equivalent, without considering nets instead of sequences in the second condition.
Many thanks in advance.
[1] "Kazhdan’s Property (T)" by B. Bekka, P. de la Harpe and A. Valette)
[2] "Amenability of discrete groups by examples"
by Kate Juschenko
operator-theory representation-theory c-star-algebras
edited Jan 13 at 23:57
Bernard
124k742117
asked Jan 13 at 23:47
OpalgalOpalgal
595
$endgroup$
Let $Gamma$ be a discrete and countable Group and let $pi:Gammato mathcal{B(H)}$ be a unitary representation.
We say that $pi$ almost has invariant vectors if for every compact (=finite) subset $FsubseteqGamma$ and $varepsilon >0$ there there exists a unit vector $xiinmathcal{H}$ such that
$$ |pi(s)xi -xi|<varepsilon; : forall sin F.$$
This is demonstrated to be equivalent to the weak containment of the trivial representation, denoted by $1_Gamma$, in $pi$ (see for example Cor. F.1.5, [1], general case of locally compact groups).
In other literature (for example Thm. A.5. and A.12, [2]) I have found the following definition:
$(pi,mathcal{H})$ admits almost invariant vectors if there exists a sequence of unit vectors $(xi_n)_nsubseteq mathcal{H}$ such that
$$|pi(s)xi_n-xi_n| to 0 ; : ; forall s in Gamma$$
which is also claimed to be equivalent to the trivial representation being weakly contained in $pi$.
Are these conditions really equivalent? If so, how would I go about understanding this equivalence? At this point I don't see how these two could be equivalent, without considering nets instead of sequences in the second condition.
Many thanks in advance.
[1] "Kazhdan’s Property (T)" by B. Bekka, P. de la Harpe and A. Valette)
[2] "Amenability of discrete groups by examples"
by Kate Juschenko
operator-theory representation-theory c-star-algebras
operator-theory representation-theory c-star-algebras
edited Jan 13 at 23:57
Bernard
124k742117
asked Jan 13 at 23:47
OpalgalOpalgal
595
edited Jan 13 at 23:57
Bernard
124k742117
asked Jan 13 at 23:47
OpalgalOpalgal
595
edited Jan 13 at 23:57
Bernard
124k742117
edited Jan 13 at 23:57
Bernard
124k742117
edited Jan 13 at 23:57
Bernard
124k742117
124k742117
asked Jan 13 at 23:47
OpalgalOpalgal
595
asked Jan 13 at 23:47
OpalgalOpalgal
595
asked Jan 13 at 23:47
OpalgalOpalgal
595
595
add a comment |
add a comment |
1 Answer
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$begingroup$
The implication "$(pi, H)$ admits almost invariant vectors" $Rightarrow$ "$(pi,H)$ almost admits invariant vectors" is relatively straightforward: Choose a sequence $(xi_n)$ in $H$ with $|pi(s)xi_n-xi_n|to 0$ for all $sinGamma$. If $FsubsetGamma$ is finite and $varepsilon>0$ is given, then for each $sin F$ there is some $N_s$ such that $|pi(s)xi_n-xi_n|<varepsilon$ for $ngeq N_s$. Now choose $ngeqmax{N_s:sin F}$, and put $xi=xi_n$.
For the other direction, let $(F_n)$ be an increasing sequence of finite subsets of $Gamma$ such that $cup_nF_n=Gamma$. For each $n$, there exists $xi_nin H$ with $|pi(s)xi_n-xi_n|<frac{1}{n}$ for all $sin F_n$. Then $(xi_n)$ is your sequence for the criterion of almost invariant vectors.
answered Jan 14 at 0:01
AweyganAweygan
14.8k21442
$endgroup$
add a comment |
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$begingroup$
The implication "$(pi, H)$ admits almost invariant vectors" $Rightarrow$ "$(pi,H)$ almost admits invariant vectors" is relatively straightforward: Choose a sequence $(xi_n)$ in $H$ with $|pi(s)xi_n-xi_n|to 0$ for all $sinGamma$. If $FsubsetGamma$ is finite and $varepsilon>0$ is given, then for each $sin F$ there is some $N_s$ such that $|pi(s)xi_n-xi_n|<varepsilon$ for $ngeq N_s$. Now choose $ngeqmax{N_s:sin F}$, and put $xi=xi_n$.
For the other direction, let $(F_n)$ be an increasing sequence of finite subsets of $Gamma$ such that $cup_nF_n=Gamma$. For each $n$, there exists $xi_nin H$ with $|pi(s)xi_n-xi_n|<frac{1}{n}$ for all $sin F_n$. Then $(xi_n)$ is your sequence for the criterion of almost invariant vectors.
answered Jan 14 at 0:01
AweyganAweygan
14.8k21442
$endgroup$
add a comment |
$begingroup$
The implication "$(pi, H)$ admits almost invariant vectors" $Rightarrow$ "$(pi,H)$ almost admits invariant vectors" is relatively straightforward: Choose a sequence $(xi_n)$ in $H$ with $|pi(s)xi_n-xi_n|to 0$ for all $sinGamma$. If $FsubsetGamma$ is finite and $varepsilon>0$ is given, then for each $sin F$ there is some $N_s$ such that $|pi(s)xi_n-xi_n|<varepsilon$ for $ngeq N_s$. Now choose $ngeqmax{N_s:sin F}$, and put $xi=xi_n$.
For the other direction, let $(F_n)$ be an increasing sequence of finite subsets of $Gamma$ such that $cup_nF_n=Gamma$. For each $n$, there exists $xi_nin H$ with $|pi(s)xi_n-xi_n|<frac{1}{n}$ for all $sin F_n$. Then $(xi_n)$ is your sequence for the criterion of almost invariant vectors.
answered Jan 14 at 0:01
AweyganAweygan
14.8k21442
$endgroup$
add a comment |
$begingroup$
The implication "$(pi, H)$ admits almost invariant vectors" $Rightarrow$ "$(pi,H)$ almost admits invariant vectors" is relatively straightforward: Choose a sequence $(xi_n)$ in $H$ with $|pi(s)xi_n-xi_n|to 0$ for all $sinGamma$. If $FsubsetGamma$ is finite and $varepsilon>0$ is given, then for each $sin F$ there is some $N_s$ such that $|pi(s)xi_n-xi_n|<varepsilon$ for $ngeq N_s$. Now choose $ngeqmax{N_s:sin F}$, and put $xi=xi_n$.
For the other direction, let $(F_n)$ be an increasing sequence of finite subsets of $Gamma$ such that $cup_nF_n=Gamma$. For each $n$, there exists $xi_nin H$ with $|pi(s)xi_n-xi_n|<frac{1}{n}$ for all $sin F_n$. Then $(xi_n)$ is your sequence for the criterion of almost invariant vectors.
answered Jan 14 at 0:01
AweyganAweygan
14.8k21442
$endgroup$
The implication "$(pi, H)$ admits almost invariant vectors" $Rightarrow$ "$(pi,H)$ almost admits invariant vectors" is relatively straightforward: Choose a sequence $(xi_n)$ in $H$ with $|pi(s)xi_n-xi_n|to 0$ for all $sinGamma$. If $FsubsetGamma$ is finite and $varepsilon>0$ is given, then for each $sin F$ there is some $N_s$ such that $|pi(s)xi_n-xi_n|<varepsilon$ for $ngeq N_s$. Now choose $ngeqmax{N_s:sin F}$, and put $xi=xi_n$.
For the other direction, let $(F_n)$ be an increasing sequence of finite subsets of $Gamma$ such that $cup_nF_n=Gamma$. For each $n$, there exists $xi_nin H$ with $|pi(s)xi_n-xi_n|<frac{1}{n}$ for all $sin F_n$. Then $(xi_n)$ is your sequence for the criterion of almost invariant vectors.
answered Jan 14 at 0:01
AweyganAweygan
14.8k21442
answered Jan 14 at 0:01
AweyganAweygan
14.8k21442
answered Jan 14 at 0:01
AweyganAweygan
14.8k21442
answered Jan 14 at 0:01
AweyganAweygan
14.8k21442
14.8k21442
add a comment |
add a comment |
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