C*-algebra without finite-dimensional representations is simple?
$begingroup$
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?
operator-theory operator-algebras c-star-algebras
asked Jan 12 at 16:35
mathrookiemathrookie
936512
$endgroup$
add a comment |
$begingroup$
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?
operator-theory operator-algebras c-star-algebras
asked Jan 12 at 16:35
mathrookiemathrookie
936512
$endgroup$
add a comment |
$begingroup$
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?
operator-theory operator-algebras c-star-algebras
asked Jan 12 at 16:35
mathrookiemathrookie
936512
$endgroup$
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
operator-theory operator-algebras c-star-algebras
asked Jan 12 at 16:35
mathrookiemathrookie
936512
asked Jan 12 at 16:35
mathrookiemathrookie
936512
edited Jan 12 at 17:21
user42761
asked Jan 12 at 16:35
mathrookiemathrookie
936512
asked Jan 12 at 16:35
mathrookiemathrookie
936512
asked Jan 12 at 16:35
mathrookiemathrookie
936512
936512
add a comment |
add a comment |
1 Answer
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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 $A_0$.
answered Jan 12 at 19:21
Martin ArgeramiMartin Argerami
129k1184185
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1 Answer
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1 Answer
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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 $A_0$.
answered Jan 12 at 19:21
Martin ArgeramiMartin Argerami
129k1184185
$endgroup$
add a comment |
$begingroup$
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 $A_0$.
answered Jan 12 at 19:21
Martin ArgeramiMartin Argerami
129k1184185
$endgroup$
add a comment |
$begingroup$
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 $A_0$.
answered Jan 12 at 19:21
Martin ArgeramiMartin Argerami
129k1184185
$endgroup$
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 $A_0$.
answered Jan 12 at 19:21
Martin ArgeramiMartin Argerami
129k1184185
answered Jan 12 at 19:21
Martin ArgeramiMartin Argerami
129k1184185
answered Jan 12 at 19:21
Martin ArgeramiMartin Argerami
129k1184185
answered Jan 12 at 19:21
Martin ArgeramiMartin Argerami
129k1184185
129k1184185
add a comment |
add a comment |
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