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I feel the following assertion is true but have no evidence to prove:

There exists an infinite dimensional C*-algebra such any cyclic representation $\pi$ of $A$ is finite dimensional! Probably $\bigoplus_{1}^{\infty}M_2(\mathbb{C})$ works.

ABB
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1 Answers1

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I think that one doesn't work.

In fact, if $A$ is any separable infinite-dimensional C$^*$-algebra, then it has a faithful state $\varphi$. If we do GNS for $\varphi$, then $\pi_\varphi$ is a cyclic representation. And it is infinite-dimensional, because $A\subset H_\varphi$ (due to the fact that $\varphi $ is faithful).

Martin Argerami
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  • Dear Martin, Thanks for your pay attention. You made an example of an infinite dimensional C*-algebra which has an infinite dimensional cyclic representation. But the question is something else. – ABB Nov 11 '15 at 10:38
  • Exactly. So "there exists a C $^*$-algebra with all its cyclic representations finite-dimensional" does not hold. How is this "something else"? – Martin Argerami Nov 11 '15 at 13:50
  • @AliBagheri Martin is not giving an example but he is proving that every separable infinite dimensional algebra admits an infinite dimensional cyclic representation. So your question has negative answer if the following statement is true: every $C^*$ algebra can be surjectively maped to a separable algebra. Is it a true statement? – Ali Taghavi Jun 13 '18 at 10:27