minimal projection in a $C^*$ algebra
If $(H,pi)$ is a nonzero representation of a $C^*$ algebra $A$,does there exist a minimal projection $Ein A$ such that $pi(E)neq 0$?
operator-theory operator-algebras c-star-algebras von-neumann-algebras
asked Nov 26 at 3:00
mathrookie
804512
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If $(H,pi)$ is a nonzero representation of a $C^*$ algebra $A$,does there exist a minimal projection $Ein A$ such that $pi(E)neq 0$?
operator-theory operator-algebras c-star-algebras von-neumann-algebras
asked Nov 26 at 3:00
mathrookie
804512
add a comment |
If $(H,pi)$ is a nonzero representation of a $C^*$ algebra $A$,does there exist a minimal projection $Ein A$ such that $pi(E)neq 0$?
operator-theory operator-algebras c-star-algebras von-neumann-algebras
asked Nov 26 at 3:00
mathrookie
804512
If $(H,pi)$ is a nonzero representation of a $C^*$ algebra $A$,does there exist a minimal projection $Ein A$ such that $pi(E)neq 0$?
operator-theory operator-algebras c-star-algebras von-neumann-algebras
operator-theory operator-algebras c-star-algebras von-neumann-algebras
asked Nov 26 at 3:00
mathrookie
804512
asked Nov 26 at 3:00
mathrookie
804512
asked Nov 26 at 3:00
mathrookie
804512
asked Nov 26 at 3:00
mathrookie
804512
asked Nov 26 at 3:00
mathrookie
804512
804512
add a comment |
add a comment |
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No. Let $A=B(H)$ for some Hilbert space $H$, let $rho:B(H)to B(H)/K(H)=Q(H)$ be the quotient map, and let $(K,pi)$ be a non-zero representation of $Q(H)$. Then $(K,picircrho)$ is a non-zero representation of $B(H)$ with $picircrho(P)=0$ for every finite-rank projection $P$ in $B(H)$, hence for all minimal projections.
answered Nov 26 at 3:04
Aweygan
13.4k21441
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1 Answer
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1 Answer
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No. Let $A=B(H)$ for some Hilbert space $H$, let $rho:B(H)to B(H)/K(H)=Q(H)$ be the quotient map, and let $(K,pi)$ be a non-zero representation of $Q(H)$. Then $(K,picircrho)$ is a non-zero representation of $B(H)$ with $picircrho(P)=0$ for every finite-rank projection $P$ in $B(H)$, hence for all minimal projections.
answered Nov 26 at 3:04
Aweygan
13.4k21441
add a comment |
No. Let $A=B(H)$ for some Hilbert space $H$, let $rho:B(H)to B(H)/K(H)=Q(H)$ be the quotient map, and let $(K,pi)$ be a non-zero representation of $Q(H)$. Then $(K,picircrho)$ is a non-zero representation of $B(H)$ with $picircrho(P)=0$ for every finite-rank projection $P$ in $B(H)$, hence for all minimal projections.
answered Nov 26 at 3:04
Aweygan
13.4k21441
add a comment |
No. Let $A=B(H)$ for some Hilbert space $H$, let $rho:B(H)to B(H)/K(H)=Q(H)$ be the quotient map, and let $(K,pi)$ be a non-zero representation of $Q(H)$. Then $(K,picircrho)$ is a non-zero representation of $B(H)$ with $picircrho(P)=0$ for every finite-rank projection $P$ in $B(H)$, hence for all minimal projections.
answered Nov 26 at 3:04
Aweygan
13.4k21441
No. Let $A=B(H)$ for some Hilbert space $H$, let $rho:B(H)to B(H)/K(H)=Q(H)$ be the quotient map, and let $(K,pi)$ be a non-zero representation of $Q(H)$. Then $(K,picircrho)$ is a non-zero representation of $B(H)$ with $picircrho(P)=0$ for every finite-rank projection $P$ in $B(H)$, hence for all minimal projections.
answered Nov 26 at 3:04
Aweygan
13.4k21441
answered Nov 26 at 3:04
Aweygan
13.4k21441
answered Nov 26 at 3:04
Aweygan
13.4k21441
answered Nov 26 at 3:04
Aweygan
13.4k21441
13.4k21441
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