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Does there exist a $C^*$-algebra $A$ such that the center of $A$ is $0$ and $A$ also has a tracial state?

I know the fact that the center of $\mathcal{K}(H)$ is $0$, but $\mathcal{K}(H)$ has no tracial states.

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  • $\begingroup$ But the center of $A$ is not 0,$(0,Id)$ is the center element. $\endgroup$ – mathrookie Jan 12 at 12:38
  • $\begingroup$ Oops, you're right, I'll delete that erroneous claim $\endgroup$ – Yemon Choi Jan 12 at 13:07
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There are lots of AF simple nonunital C$^*$-algebras with (finite) traces (these have trivial centre). For example, begin with the usual $2^{\infty}$ UHF algebra, call it $B$, and take an infinite strictly increasing sequence of projections, $(p_n)$, and form $\cup_n p_n B p_n$; let $A$ be its closure. Then $A$ is simple (easy to check); it is nonunital (very easy) [and being simple, is thus centreless]; it has unique trace (given by the restriction of the unique trace on $B$) (easy). $A$ is of course a hereditary subalgebra of $B$, and there are hordes of similar examples.

If we set $\alpha$ to be the limit of the traces of the $p_n$, then there is a significant difference (certainly wrt classification) between the algebras we obtain when $\alpha$ is rational, and when $\alpha $ is irrational.

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  • $\begingroup$ Suppose $\tau$ is the trace of $B$,how to check that $\tau|_{A}$ is nonzero? $\endgroup$ – mathrookie Jan 12 at 16:30
  • $\begingroup$ Since $B$ is simple, $\tau (aa*) >0$ for all $a \in B$ (since the set of $a \in B$ st $\tau (aa*) = 0$ is an ideal of $B$), hence for $a \in A$. $\endgroup$ – David Handelman Jan 12 at 19:16
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    $\begingroup$ @mathrookie Of course, I meant "for all nonzero $a \in B$"; I wish we could edit comments. $\endgroup$ – David Handelman Jan 12 at 19:27
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I think you can obtain an example by modifying the classical Toeplitz algebra. $\newcommand{\H}{\mathbf{H}}$ $\newcommand{\bT}{\mathbf{T}}$ $\newcommand{\cT}{\mathcal T}$ $\newcommand{\KH}{{\mathcal K}(\H)}$ $\newcommand{\BH}{{\mathcal B}(\H)}$

Recall that ${\mathcal T} \subseteq \BH$ where $\H=\ell^2({\mathbf N}_0))$ and $\cT/\KH\cong C(\bT)$. Let $q: \cT \to C(\bT)$ be the corresponding quotient homomorphism, let $J=\{ f\in C(\bT) \colon f(1)=0\}$ and let $\cT_0= q^{-1}(J)$. More concretely, $\cT_0$ is the set of closed algebra generated by all Toeplitz operators $T_f$ for which the symbol function $f$ belongs to $J$.

Since $\cT_0$ quotients onto the commutative ${\rm C}^*$-algebra $J$ it has loads of tracial states (just pull back the states on $J$).

On the other hand, suppose $a\in Z(\cT)$. Then $[a,k]=0$ for all $k\in \KH$. But it is standard and not too difficult to show that if $b\in\BH$ satisfies $[k,b]=0$ for all $k\in\KH$, then $b$ is a scalar multiple of the identity. Since $q(I_\H)\notin J$ the only possibility is that $a=0$.

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  • $\begingroup$ Nice answer. But the statement that $\mathcal{T}_0$ consists of Toeplitz operators whose symbol vanishes at $1$; you need to throw in the compact perturbations or allow products. $\endgroup$ – Mateusz Wasilewski Jan 12 at 14:46
  • $\begingroup$ Irrational rotation algebras (quantum tori) are another example. $\endgroup$ – Nik Weaver Jan 12 at 14:56
  • $\begingroup$ Also, the reduced group C*-algebra of any discrete group always has a tracial state. $\endgroup$ – Nik Weaver Jan 12 at 15:03
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    $\begingroup$ Or just $A = M_n$ ... $\endgroup$ – Nik Weaver Jan 12 at 15:03
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    $\begingroup$ Thanks Mateusz. @NikWeaver: the OP wants $Z(A)=0$ not $\dim Z(A)=1$ $\endgroup$ – Yemon Choi Jan 12 at 15:51
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Jacelon's $\mathcal W$ (https://arxiv.org/abs/1006.5397) should also do.

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