C*-algebra without finite-dimensional representations is simple?

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Suppose $A$ is an infinite dimensional simple $C^*$-algebra. Then it has no non-zero finite dimensional representations. Is the converse also true? That is to say, if a $C^*$-algebra has no finite dimensional representation, can we conclude that the $C^*$-algebra is simple?










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    Suppose $A$ is an infinite dimensional simple $C^*$-algebra. Then it has no non-zero finite dimensional representations. Is the converse also true? That is to say, if a $C^*$-algebra has no finite dimensional representation, can we conclude that the $C^*$-algebra is simple?










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      $begingroup$


      Suppose $A$ is an infinite dimensional simple $C^*$-algebra. Then it has no non-zero finite dimensional representations. Is the converse also true? That is to say, if a $C^*$-algebra has no finite dimensional representation, can we conclude that the $C^*$-algebra is simple?










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      Suppose $A$ is an infinite dimensional simple $C^*$-algebra. Then it has no non-zero finite dimensional representations. Is the converse also true? That is to say, if a $C^*$-algebra has no finite dimensional representation, can we conclude that the $C^*$-algebra is simple?







      operator-theory operator-algebras c-star-algebras






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      edited Jan 12 at 17:21







      user42761

















      asked Jan 12 at 16:35









      mathrookiemathrookie

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          Take any simple infinite-dimensional $A_0$, and form $A=A_0oplus A_0$. Then $A$ is not simple. And $A$ has no finite-dimensional representations, because all representations restrict to representations of $A_0$.






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            $begingroup$

            Take any simple infinite-dimensional $A_0$, and form $A=A_0oplus A_0$. Then $A$ is not simple. And $A$ has no finite-dimensional representations, because all representations restrict to representations of $A_0$.






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              1












              $begingroup$

              Take any simple infinite-dimensional $A_0$, and form $A=A_0oplus A_0$. Then $A$ is not simple. And $A$ has no finite-dimensional representations, because all representations restrict to representations of $A_0$.






              share|cite|improve this answer









              $endgroup$
















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                $begingroup$

                Take any simple infinite-dimensional $A_0$, and form $A=A_0oplus A_0$. Then $A$ is not simple. And $A$ has no finite-dimensional representations, because all representations restrict to representations of $A_0$.






                share|cite|improve this answer









                $endgroup$



                Take any simple infinite-dimensional $A_0$, and form $A=A_0oplus A_0$. Then $A$ is not simple. And $A$ has no finite-dimensional representations, because all representations restrict to representations of $A_0$.







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                answered Jan 12 at 19:21









                Martin ArgeramiMartin Argerami

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