$*$ homomorphism $phi$ from $A$ to multiplier algebra $M(I)$

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If $I$ is a closed ideal in $C^*$ algebra $A$,then there is a unique $*$ homomorphism $phi$ from $A$ to $M(I)$ which extends the $*$ homomorphism $Irightarrow M(I),$where $M(I)$ is the multiplier algebra of $I$.



My question is:If $A$ has a unit $e$,the map $phi$ must not be zero since $e mapsto (L_e,R_e)$.



If $A$ is not unital,can we conclude that $phi$ is also a nonzero $*$ homomorphism?




operator-theory operator-algebras c-star-algebras




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If $I$ is a closed ideal in $C^*$ algebra $A$,then there is a unique $*$ homomorphism $phi$ from $A$ to $M(I)$ which extends the $*$ homomorphism $Irightarrow M(I),$where $M(I)$ is the multiplier algebra of $I$.



My question is:If $A$ has a unit $e$,the map $phi$ must not be zero since $e mapsto (L_e,R_e)$.



If $A$ is not unital,can we conclude that $phi$ is also a nonzero $*$ homomorphism?




operator-theory operator-algebras c-star-algebras




share|cite|improve this question















  • 1




    Looking at the quantity of questions you post here, I suggest you to think more about them yourself and try to build knowledge.
    – André S.
    Jul 25 at 11:57












up vote
1
down vote

favorite









up vote
1
down vote

favorite











If $I$ is a closed ideal in $C^*$ algebra $A$,then there is a unique $*$ homomorphism $phi$ from $A$ to $M(I)$ which extends the $*$ homomorphism $Irightarrow M(I),$where $M(I)$ is the multiplier algebra of $I$.



My question is:If $A$ has a unit $e$,the map $phi$ must not be zero since $e mapsto (L_e,R_e)$.



If $A$ is not unital,can we conclude that $phi$ is also a nonzero $*$ homomorphism?




operator-theory operator-algebras c-star-algebras




share|cite|improve this question











If $I$ is a closed ideal in $C^*$ algebra $A$,then there is a unique $*$ homomorphism $phi$ from $A$ to $M(I)$ which extends the $*$ homomorphism $Irightarrow M(I),$where $M(I)$ is the multiplier algebra of $I$.



My question is:If $A$ has a unit $e$,the map $phi$ must not be zero since $e mapsto (L_e,R_e)$.



If $A$ is not unital,can we conclude that $phi$ is also a nonzero $*$ homomorphism?





operator-theory operator-algebras c-star-algebras





share|cite|improve this question










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asked Jul 25 at 7:14









mathrookie

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  • 1




    Looking at the quantity of questions you post here, I suggest you to think more about them yourself and try to build knowledge.
    – André S.
    Jul 25 at 11:57












  • 1




    Looking at the quantity of questions you post here, I suggest you to think more about them yourself and try to build knowledge.
    – André S.
    Jul 25 at 11:57







1




1




Looking at the quantity of questions you post here, I suggest you to think more about them yourself and try to build knowledge.
– André S.
Jul 25 at 11:57




Looking at the quantity of questions you post here, I suggest you to think more about them yourself and try to build knowledge.
– André S.
Jul 25 at 11:57










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If $phi$ is zero, then also $I to M(I)$






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    If $phi$ is zero, then also $I to M(I)$






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      If $phi$ is zero, then also $I to M(I)$






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        If $phi$ is zero, then also $I to M(I)$






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        If $phi$ is zero, then also $I to M(I)$







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        edited Jul 28 at 10:29


























        answered Jul 25 at 12:00









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