Mixing
Normal u.c.p extension of Schur-multiplier
$begingroup$
I'm struggeling with the proof of a theorem in [BO08]. The first part before the line is what I think I understood. The part after that I don't understand at all.
Let $Gamma$ be a discrete group and $mathbb{C}[Gamma]$ the group ring of $Gamma$. Let $C_lambda^ast(Gamma)$ be the reduced group $C^ast$-algebra, i.e. the completion of $mathbb{C}[Gamma]$ wrt. the norm $|x|=|lambda(x)|$ where λ is the left regular representation of $Gamma$ on $mathbb{B}(ell^2(Gamma)$. Let $$L(Gamma) = C_lambda^ast(Gamma)'' subset mathcal{B}(ell^2(Gamma))$$ be the so called group von Neumann algebra. The full group $C^ast$-algebra is defined as $C^*(Gamma) = overline{mathbb{C}(Gamma)}^{|cdot|_u}$ with $| x|_u = sup{ |pi(x)| ; | ; pi text{ is a unitary representation of } Gamma}$.
For two $C^ast$-algebras $A$ and $B$ we say that a map $varphi_Ato B$ is u.c.p if $varphi$ is unital and $varphi_n:M_n(A)→M_n(B)$ defined by $varphi_n([a_{i,j}])=[varphi(a_{i,j})]$ maps positive matrices to positive matrices for all $nin mathbb{N}$.
$varphi$ is called normal if it is continuous wrt. ultra-weak topology.
A function $varphi:Gammato mathbb{C}$ is called positive definite if for every finite subset $F:={s_1, dots, s_n}subseteq Gamma$ the matrix
$$[varphi(s_i^{-1}s_j)]_{i,j}in M_n(mathbb{C}) $$
is positive semidefinite.
For a function $varphi:Gammatomathbb{C}$ set
begin{equation*}
begin{split}
w_varphi: mathbb{C}[Gamma] &to mathbb{C}\
sum_{tinGamma}alpha_t t &mapsto sum_{tinGamma} alpha_t varphi(t)
end{split}
end{equation*}
and
begin{equation*}
begin{split}
m_varphi: mathbb{C}[Gamma] &to mathbb{C}[Gamma]\
sum_{tinGamma}alpha_t t &mapsto sum_{tinGamma} alpha_t varphi(t)t.
end{split}
end{equation*}
The map $m_varphi$ is also called the Schur-multiplier.
Theorem(2.5.11, [BO08])Let $varphi:Gammatomathbb{C}$ be a function with $varphi(e) =1$. The following conditions are equivalent:
- The functional $w_varphi$ extends to a positive functional on $C^ast(Gamma)$.
- The Schur multiplier $m_varphi$ extends to a u.c.p. map on the two group $C^ast$-algebras $C^ast(Gamma)$ and $C^ast_lambda (Gamma)$ and to a normal u.c.p map on $L(Gamma)$.
Proof:Since $w_varphi$ extends to a functional on $C^ast(Gamma)$ it follows that $w_varphi$ extends to a state on $C^ast(Gamma)$, which will also be denoted by $w_varphi$. Embedding $C^ast(Gamma)subseteq mathcal{B}(H_u)$ where $H_u$ is the Hilbert space of the universal representation $(pi_u, H_u)$ of $C^ast(Gamma)$, we can regard $w_varphi$ as a state on $mathcal{B}(H_u)$.
We define a map
begin{equation*}
begin{split}
phi: C_lambda^ast(Gamma)otimes 1 cong C_lambda^ast(Gamma) &to C_lambda^ast(Gamma)otimes C^ast(Gamma) subseteq C_lambda^ast(Gamma)otimes mathcal{B}(H_u)\
sum_{tinGamma}alpha_tlambda(t)otimes 1 &mapsto sum_{tinGamma}alpha_t(lambda(t)otimes t)
end{split}
end{equation*}
which is a $ast$-homomorphism: By Fell's absorption there is a unitary operator $U$ such that $U^ast(lambda otimes pi_u)(x)U = (lambdaotimes 1)(x)$ for $xin mathbb{C}[Gamma]$ with
begin{equation*}
begin{split}
|phi(sum_{tin Gamma} alpha_t (lambda(t)otimes 1))| &= |sum_{tin Gamma} alpha_t (lambda(t)otimes t)| = |(lambdaotimes pi_u)(sum_{tin Gamma} alpha_t t)|\
&=|U^ast (lambdaotimes1)(sum_{tinGamma} alpha_t t)U| = |sum_{tinGamma} alpha_t(lambda(t)otimes 1)|.
end{split}
end{equation*}
We conclude that $phi$ extends to a normal $ast$-homomorphism $phi:L(Gamma) to L(Gamma)overline{otimes} B(H_u)$(Why?).
Checking that $(id_{L[Gamma)} otimes w_varphi)circ phi$ coincides with $m_varphi$ (identifying $L(Gamma)overline{otimes} 1 cong L(Gamma)$) concludes the group von Neumann algebra case. The reduced group $C^ast$-algebra case follows directly.(Why?)
I would be very thankful if someone could give me a hint or some explanations. Thank you very much in advance!
[BO08] Brown and Ozawa, "C∗-Algebras and Fnite-Dimensional Approximations"
operator-algebras c-star-algebras von-neumann-algebras
asked Jan 18 at 18:30
OpalgalOpalgal
595
$endgroup$
add a comment |
$begingroup$
I'm struggeling with the proof of a theorem in [BO08]. The first part before the line is what I think I understood. The part after that I don't understand at all.
Let $Gamma$ be a discrete group and $mathbb{C}[Gamma]$ the group ring of $Gamma$. Let $C_lambda^ast(Gamma)$ be the reduced group $C^ast$-algebra, i.e. the completion of $mathbb{C}[Gamma]$ wrt. the norm $|x|=|lambda(x)|$ where λ is the left regular representation of $Gamma$ on $mathbb{B}(ell^2(Gamma)$. Let $$L(Gamma) = C_lambda^ast(Gamma)'' subset mathcal{B}(ell^2(Gamma))$$ be the so called group von Neumann algebra. The full group $C^ast$-algebra is defined as $C^*(Gamma) = overline{mathbb{C}(Gamma)}^{|cdot|_u}$ with $| x|_u = sup{ |pi(x)| ; | ; pi text{ is a unitary representation of } Gamma}$.
For two $C^ast$-algebras $A$ and $B$ we say that a map $varphi_Ato B$ is u.c.p if $varphi$ is unital and $varphi_n:M_n(A)→M_n(B)$ defined by $varphi_n([a_{i,j}])=[varphi(a_{i,j})]$ maps positive matrices to positive matrices for all $nin mathbb{N}$.
$varphi$ is called normal if it is continuous wrt. ultra-weak topology.
A function $varphi:Gammato mathbb{C}$ is called positive definite if for every finite subset $F:={s_1, dots, s_n}subseteq Gamma$ the matrix
$$[varphi(s_i^{-1}s_j)]_{i,j}in M_n(mathbb{C}) $$
is positive semidefinite.
For a function $varphi:Gammatomathbb{C}$ set
begin{equation*}
begin{split}
w_varphi: mathbb{C}[Gamma] &to mathbb{C}\
sum_{tinGamma}alpha_t t &mapsto sum_{tinGamma} alpha_t varphi(t)
end{split}
end{equation*}
and
begin{equation*}
begin{split}
m_varphi: mathbb{C}[Gamma] &to mathbb{C}[Gamma]\
sum_{tinGamma}alpha_t t &mapsto sum_{tinGamma} alpha_t varphi(t)t.
end{split}
end{equation*}
The map $m_varphi$ is also called the Schur-multiplier.
Theorem(2.5.11, [BO08])Let $varphi:Gammatomathbb{C}$ be a function with $varphi(e) =1$. The following conditions are equivalent:
- The functional $w_varphi$ extends to a positive functional on $C^ast(Gamma)$.
- The Schur multiplier $m_varphi$ extends to a u.c.p. map on the two group $C^ast$-algebras $C^ast(Gamma)$ and $C^ast_lambda (Gamma)$ and to a normal u.c.p map on $L(Gamma)$.
Proof:Since $w_varphi$ extends to a functional on $C^ast(Gamma)$ it follows that $w_varphi$ extends to a state on $C^ast(Gamma)$, which will also be denoted by $w_varphi$. Embedding $C^ast(Gamma)subseteq mathcal{B}(H_u)$ where $H_u$ is the Hilbert space of the universal representation $(pi_u, H_u)$ of $C^ast(Gamma)$, we can regard $w_varphi$ as a state on $mathcal{B}(H_u)$.
We define a map
begin{equation*}
begin{split}
phi: C_lambda^ast(Gamma)otimes 1 cong C_lambda^ast(Gamma) &to C_lambda^ast(Gamma)otimes C^ast(Gamma) subseteq C_lambda^ast(Gamma)otimes mathcal{B}(H_u)\
sum_{tinGamma}alpha_tlambda(t)otimes 1 &mapsto sum_{tinGamma}alpha_t(lambda(t)otimes t)
end{split}
end{equation*}
which is a $ast$-homomorphism: By Fell's absorption there is a unitary operator $U$ such that $U^ast(lambda otimes pi_u)(x)U = (lambdaotimes 1)(x)$ for $xin mathbb{C}[Gamma]$ with
begin{equation*}
begin{split}
|phi(sum_{tin Gamma} alpha_t (lambda(t)otimes 1))| &= |sum_{tin Gamma} alpha_t (lambda(t)otimes t)| = |(lambdaotimes pi_u)(sum_{tin Gamma} alpha_t t)|\
&=|U^ast (lambdaotimes1)(sum_{tinGamma} alpha_t t)U| = |sum_{tinGamma} alpha_t(lambda(t)otimes 1)|.
end{split}
end{equation*}
We conclude that $phi$ extends to a normal $ast$-homomorphism $phi:L(Gamma) to L(Gamma)overline{otimes} B(H_u)$(Why?).
Checking that $(id_{L[Gamma)} otimes w_varphi)circ phi$ coincides with $m_varphi$ (identifying $L(Gamma)overline{otimes} 1 cong L(Gamma)$) concludes the group von Neumann algebra case. The reduced group $C^ast$-algebra case follows directly.(Why?)
I would be very thankful if someone could give me a hint or some explanations. Thank you very much in advance!
[BO08] Brown and Ozawa, "C∗-Algebras and Fnite-Dimensional Approximations"
operator-algebras c-star-algebras von-neumann-algebras
asked Jan 18 at 18:30
OpalgalOpalgal
595
$endgroup$
add a comment |
$begingroup$
I'm struggeling with the proof of a theorem in [BO08]. The first part before the line is what I think I understood. The part after that I don't understand at all.
Let $Gamma$ be a discrete group and $mathbb{C}[Gamma]$ the group ring of $Gamma$. Let $C_lambda^ast(Gamma)$ be the reduced group $C^ast$-algebra, i.e. the completion of $mathbb{C}[Gamma]$ wrt. the norm $|x|=|lambda(x)|$ where λ is the left regular representation of $Gamma$ on $mathbb{B}(ell^2(Gamma)$. Let $$L(Gamma) = C_lambda^ast(Gamma)'' subset mathcal{B}(ell^2(Gamma))$$ be the so called group von Neumann algebra. The full group $C^ast$-algebra is defined as $C^*(Gamma) = overline{mathbb{C}(Gamma)}^{|cdot|_u}$ with $| x|_u = sup{ |pi(x)| ; | ; pi text{ is a unitary representation of } Gamma}$.
For two $C^ast$-algebras $A$ and $B$ we say that a map $varphi_Ato B$ is u.c.p if $varphi$ is unital and $varphi_n:M_n(A)→M_n(B)$ defined by $varphi_n([a_{i,j}])=[varphi(a_{i,j})]$ maps positive matrices to positive matrices for all $nin mathbb{N}$.
$varphi$ is called normal if it is continuous wrt. ultra-weak topology.
A function $varphi:Gammato mathbb{C}$ is called positive definite if for every finite subset $F:={s_1, dots, s_n}subseteq Gamma$ the matrix
$$[varphi(s_i^{-1}s_j)]_{i,j}in M_n(mathbb{C}) $$
is positive semidefinite.
For a function $varphi:Gammatomathbb{C}$ set
begin{equation*}
begin{split}
w_varphi: mathbb{C}[Gamma] &to mathbb{C}\
sum_{tinGamma}alpha_t t &mapsto sum_{tinGamma} alpha_t varphi(t)
end{split}
end{equation*}
and
begin{equation*}
begin{split}
m_varphi: mathbb{C}[Gamma] &to mathbb{C}[Gamma]\
sum_{tinGamma}alpha_t t &mapsto sum_{tinGamma} alpha_t varphi(t)t.
end{split}
end{equation*}
The map $m_varphi$ is also called the Schur-multiplier.
Theorem(2.5.11, [BO08])Let $varphi:Gammatomathbb{C}$ be a function with $varphi(e) =1$. The following conditions are equivalent:
- The functional $w_varphi$ extends to a positive functional on $C^ast(Gamma)$.
- The Schur multiplier $m_varphi$ extends to a u.c.p. map on the two group $C^ast$-algebras $C^ast(Gamma)$ and $C^ast_lambda (Gamma)$ and to a normal u.c.p map on $L(Gamma)$.
Proof:Since $w_varphi$ extends to a functional on $C^ast(Gamma)$ it follows that $w_varphi$ extends to a state on $C^ast(Gamma)$, which will also be denoted by $w_varphi$. Embedding $C^ast(Gamma)subseteq mathcal{B}(H_u)$ where $H_u$ is the Hilbert space of the universal representation $(pi_u, H_u)$ of $C^ast(Gamma)$, we can regard $w_varphi$ as a state on $mathcal{B}(H_u)$.
We define a map
begin{equation*}
begin{split}
phi: C_lambda^ast(Gamma)otimes 1 cong C_lambda^ast(Gamma) &to C_lambda^ast(Gamma)otimes C^ast(Gamma) subseteq C_lambda^ast(Gamma)otimes mathcal{B}(H_u)\
sum_{tinGamma}alpha_tlambda(t)otimes 1 &mapsto sum_{tinGamma}alpha_t(lambda(t)otimes t)
end{split}
end{equation*}
which is a $ast$-homomorphism: By Fell's absorption there is a unitary operator $U$ such that $U^ast(lambda otimes pi_u)(x)U = (lambdaotimes 1)(x)$ for $xin mathbb{C}[Gamma]$ with
begin{equation*}
begin{split}
|phi(sum_{tin Gamma} alpha_t (lambda(t)otimes 1))| &= |sum_{tin Gamma} alpha_t (lambda(t)otimes t)| = |(lambdaotimes pi_u)(sum_{tin Gamma} alpha_t t)|\
&=|U^ast (lambdaotimes1)(sum_{tinGamma} alpha_t t)U| = |sum_{tinGamma} alpha_t(lambda(t)otimes 1)|.
end{split}
end{equation*}
We conclude that $phi$ extends to a normal $ast$-homomorphism $phi:L(Gamma) to L(Gamma)overline{otimes} B(H_u)$(Why?).
Checking that $(id_{L[Gamma)} otimes w_varphi)circ phi$ coincides with $m_varphi$ (identifying $L(Gamma)overline{otimes} 1 cong L(Gamma)$) concludes the group von Neumann algebra case. The reduced group $C^ast$-algebra case follows directly.(Why?)
I would be very thankful if someone could give me a hint or some explanations. Thank you very much in advance!
[BO08] Brown and Ozawa, "C∗-Algebras and Fnite-Dimensional Approximations"
operator-algebras c-star-algebras von-neumann-algebras
asked Jan 18 at 18:30
OpalgalOpalgal
595
$endgroup$
I'm struggeling with the proof of a theorem in [BO08]. The first part before the line is what I think I understood. The part after that I don't understand at all.
Let $Gamma$ be a discrete group and $mathbb{C}[Gamma]$ the group ring of $Gamma$. Let $C_lambda^ast(Gamma)$ be the reduced group $C^ast$-algebra, i.e. the completion of $mathbb{C}[Gamma]$ wrt. the norm $|x|=|lambda(x)|$ where λ is the left regular representation of $Gamma$ on $mathbb{B}(ell^2(Gamma)$. Let $$L(Gamma) = C_lambda^ast(Gamma)'' subset mathcal{B}(ell^2(Gamma))$$ be the so called group von Neumann algebra. The full group $C^ast$-algebra is defined as $C^*(Gamma) = overline{mathbb{C}(Gamma)}^{|cdot|_u}$ with $| x|_u = sup{ |pi(x)| ; | ; pi text{ is a unitary representation of } Gamma}$.
For two $C^ast$-algebras $A$ and $B$ we say that a map $varphi_Ato B$ is u.c.p if $varphi$ is unital and $varphi_n:M_n(A)→M_n(B)$ defined by $varphi_n([a_{i,j}])=[varphi(a_{i,j})]$ maps positive matrices to positive matrices for all $nin mathbb{N}$.
$varphi$ is called normal if it is continuous wrt. ultra-weak topology.
A function $varphi:Gammato mathbb{C}$ is called positive definite if for every finite subset $F:={s_1, dots, s_n}subseteq Gamma$ the matrix
$$[varphi(s_i^{-1}s_j)]_{i,j}in M_n(mathbb{C}) $$
is positive semidefinite.
For a function $varphi:Gammatomathbb{C}$ set
begin{equation*}
begin{split}
w_varphi: mathbb{C}[Gamma] &to mathbb{C}\
sum_{tinGamma}alpha_t t &mapsto sum_{tinGamma} alpha_t varphi(t)
end{split}
end{equation*}
and
begin{equation*}
begin{split}
m_varphi: mathbb{C}[Gamma] &to mathbb{C}[Gamma]\
sum_{tinGamma}alpha_t t &mapsto sum_{tinGamma} alpha_t varphi(t)t.
end{split}
end{equation*}
The map $m_varphi$ is also called the Schur-multiplier.
Theorem(2.5.11, [BO08])Let $varphi:Gammatomathbb{C}$ be a function with $varphi(e) =1$. The following conditions are equivalent:
- The functional $w_varphi$ extends to a positive functional on $C^ast(Gamma)$.
- The Schur multiplier $m_varphi$ extends to a u.c.p. map on the two group $C^ast$-algebras $C^ast(Gamma)$ and $C^ast_lambda (Gamma)$ and to a normal u.c.p map on $L(Gamma)$.
Proof:Since $w_varphi$ extends to a functional on $C^ast(Gamma)$ it follows that $w_varphi$ extends to a state on $C^ast(Gamma)$, which will also be denoted by $w_varphi$. Embedding $C^ast(Gamma)subseteq mathcal{B}(H_u)$ where $H_u$ is the Hilbert space of the universal representation $(pi_u, H_u)$ of $C^ast(Gamma)$, we can regard $w_varphi$ as a state on $mathcal{B}(H_u)$.
We define a map
begin{equation*}
begin{split}
phi: C_lambda^ast(Gamma)otimes 1 cong C_lambda^ast(Gamma) &to C_lambda^ast(Gamma)otimes C^ast(Gamma) subseteq C_lambda^ast(Gamma)otimes mathcal{B}(H_u)\
sum_{tinGamma}alpha_tlambda(t)otimes 1 &mapsto sum_{tinGamma}alpha_t(lambda(t)otimes t)
end{split}
end{equation*}
which is a $ast$-homomorphism: By Fell's absorption there is a unitary operator $U$ such that $U^ast(lambda otimes pi_u)(x)U = (lambdaotimes 1)(x)$ for $xin mathbb{C}[Gamma]$ with
begin{equation*}
begin{split}
|phi(sum_{tin Gamma} alpha_t (lambda(t)otimes 1))| &= |sum_{tin Gamma} alpha_t (lambda(t)otimes t)| = |(lambdaotimes pi_u)(sum_{tin Gamma} alpha_t t)|\
&=|U^ast (lambdaotimes1)(sum_{tinGamma} alpha_t t)U| = |sum_{tinGamma} alpha_t(lambda(t)otimes 1)|.
end{split}
end{equation*}
We conclude that $phi$ extends to a normal $ast$-homomorphism $phi:L(Gamma) to L(Gamma)overline{otimes} B(H_u)$(Why?).
Checking that $(id_{L[Gamma)} otimes w_varphi)circ phi$ coincides with $m_varphi$ (identifying $L(Gamma)overline{otimes} 1 cong L(Gamma)$) concludes the group von Neumann algebra case. The reduced group $C^ast$-algebra case follows directly.(Why?)
I would be very thankful if someone could give me a hint or some explanations. Thank you very much in advance!
[BO08] Brown and Ozawa, "C∗-Algebras and Fnite-Dimensional Approximations"
operator-algebras c-star-algebras von-neumann-algebras
operator-algebras c-star-algebras von-neumann-algebras
asked Jan 18 at 18:30
OpalgalOpalgal
595
asked Jan 18 at 18:30
OpalgalOpalgal
595
edited Jan 18 at 21:15
Opalgal
asked Jan 18 at 18:30
OpalgalOpalgal
595
asked Jan 18 at 18:30
OpalgalOpalgal
595
asked Jan 18 at 18:30
OpalgalOpalgal
595
595
add a comment |
add a comment |
1 Answer
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$begingroup$
It says in [BO08] that "Fell's principle is spatially implemented". That is,
$$
phi:sum_talpha_tlambda_tlongmapsto sum_talpha_t(lambda_totimes1)=U^*left(sum_talpha_t(lambda_totimes pi_u)right),Ulongmapsto sum_talpha_t(lambda_totimes pi_u).
$$
As mentioned in the book, both maps in the composition above are spatial, so they are sot continuous; so in particular $phi$ is normal.
The argument given in book works for the reduced C$^*$-algebra case if you just don't extend to $L(Gamma)$.
answered Jan 19 at 1:10
Martin ArgeramiMartin Argerami
127k1182183
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add a comment |
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$begingroup$
It says in [BO08] that "Fell's principle is spatially implemented". That is,
$$
phi:sum_talpha_tlambda_tlongmapsto sum_talpha_t(lambda_totimes1)=U^*left(sum_talpha_t(lambda_totimes pi_u)right),Ulongmapsto sum_talpha_t(lambda_totimes pi_u).
$$
As mentioned in the book, both maps in the composition above are spatial, so they are sot continuous; so in particular $phi$ is normal.
The argument given in book works for the reduced C$^*$-algebra case if you just don't extend to $L(Gamma)$.
answered Jan 19 at 1:10
Martin ArgeramiMartin Argerami
127k1182183
$endgroup$
add a comment |
$begingroup$
It says in [BO08] that "Fell's principle is spatially implemented". That is,
$$
phi:sum_talpha_tlambda_tlongmapsto sum_talpha_t(lambda_totimes1)=U^*left(sum_talpha_t(lambda_totimes pi_u)right),Ulongmapsto sum_talpha_t(lambda_totimes pi_u).
$$
As mentioned in the book, both maps in the composition above are spatial, so they are sot continuous; so in particular $phi$ is normal.
The argument given in book works for the reduced C$^*$-algebra case if you just don't extend to $L(Gamma)$.
answered Jan 19 at 1:10
Martin ArgeramiMartin Argerami
127k1182183
$endgroup$
add a comment |
$begingroup$
It says in [BO08] that "Fell's principle is spatially implemented". That is,
$$
phi:sum_talpha_tlambda_tlongmapsto sum_talpha_t(lambda_totimes1)=U^*left(sum_talpha_t(lambda_totimes pi_u)right),Ulongmapsto sum_talpha_t(lambda_totimes pi_u).
$$
As mentioned in the book, both maps in the composition above are spatial, so they are sot continuous; so in particular $phi$ is normal.
The argument given in book works for the reduced C$^*$-algebra case if you just don't extend to $L(Gamma)$.
answered Jan 19 at 1:10
Martin ArgeramiMartin Argerami
127k1182183
$endgroup$
It says in [BO08] that "Fell's principle is spatially implemented". That is,
$$
phi:sum_talpha_tlambda_tlongmapsto sum_talpha_t(lambda_totimes1)=U^*left(sum_talpha_t(lambda_totimes pi_u)right),Ulongmapsto sum_talpha_t(lambda_totimes pi_u).
$$
As mentioned in the book, both maps in the composition above are spatial, so they are sot continuous; so in particular $phi$ is normal.
The argument given in book works for the reduced C$^*$-algebra case if you just don't extend to $L(Gamma)$.
answered Jan 19 at 1:10
Martin ArgeramiMartin Argerami
127k1182183
answered Jan 19 at 1:10
Martin ArgeramiMartin Argerami
127k1182183
answered Jan 19 at 1:10
Martin ArgeramiMartin Argerami
127k1182183
answered Jan 19 at 1:10
Martin ArgeramiMartin Argerami
127k1182183
127k1182183
add a comment |
add a comment |
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Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
