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If $A$ is a finite dimensional simple $C^*$ algebra,then it has the form of $M_n(\mathbb{C})$,which has unique tracial state.

My question is:If $A$ is an unital infinite dimensional simple $C^*$ algebra,what is the structure of $A$?Does it have a relationship with matrix algebras?I cannot think of a concrete example (unital infinite dimensional simple $C^*$ algebra whose tracial state exists).

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For a "matrix-like" example, you can take UHF$(2^\infty)$. It's simple and it has a unique tracial state.

But there are lots of examples of different kinds. If you take any infinite discrete group $G$, you can form $C_r^*(G)$, the reduced C$^*$-algebra (i.e., the C$^*$-algebra generated by the left-regular representation). For any $G$, this always has a faithful trace, namely $$\tau(x)=\langle xe,e\rangle,$$ where $e$ is the vector induced by the unit of $G$. And, for many groups, $C_r^*(G)$ is simple. The free groups, for instance So $C_r^*(\mathbb F_n)$, $n\in\mathbb N$ are examples of infinite-dimensional, simple C$^*$-algebras with a faithful trace.

Among many, $G$ above could be any of

  • $\mathbb F_n$, $n\in\mathbb N$
  • PSL$_n(\mathbb Z)$, $n\geq2$
  • non-trivial free products
  • non-solvable subgroups of PSL$_2(\mathbb R)$
  • torsion-free non-elementary Gromov hyperbolic groups
  • any Zariski-dense subgroup with with centre reduced to $\{1\}$ in a connected semi-simple real Lie group without compact factor
  • centerless mapping class groups and outer automorphism groups of free groups
  • irreducible Coxeter groups which are neither finite nor affine
  • Tarski monster groups
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  • $\begingroup$ Not necessarily, but it is in many cases. $\endgroup$ – Martin Argerami Nov 2 '18 at 15:05
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    $\begingroup$ In $\ell^\infty$, there are uncountably many pairwise orthogonal infinite projections. These pass to the quotient. Any C$^*$-algebra with uncountable many pairwise orthogonal projections cannot have a faithful state. You would have, for any finite set $F$, $$1=\varphi(1)\geq\varphi(\sum_{ j\in F}p_j)=\sum_{j\in F}\varphi(p_j).$$ It follows that $\sum_j\varphi(p_j)<\infty$, but this requires only countably many to be nonzero. $\endgroup$ – Martin Argerami Nov 2 '18 at 16:38

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