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\begin{document}
\title{Set theory and C* algebras}
\author{Edited by }

\urlstub{settheorycstar}

\maketitle




\section{Ultrapowers}





\begin{problemblock}

\begin{problem}[.1] Does every separable C*-algebra embed into an ultrapower of
$\mathcal O_2$ with respect to an ultrafilter on $\mathbb N$? \end{problem}




\begin{remark}
\by{Kirchberg}Every exact C*-algebra embeds into $\mathcal O_2$.\end{remark}

\end{problemblock}



\begin{problemblock}If $\phi_j\colon A\to \prod_{\mathcal U} \mathcal O_2$ are *-homomorphisms, write $\phi_1\leq
\phi_2$ if there is a partial isometry $u$ in $A$ such that $u^*\phi_2
u=\phi_1$.

\begin{problem}[.2] Assume $A$ is separable and  $A$ embeds into
$\prod_{\mathcal U}\mathcal O_2$. Is there a $\leq$-maximal embedding $\phi$ of $A$ into
$\prod_{\mathcal U}\mathcal O_2$? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.3] Can one prove in ZFC that for some free ultrafilter $\mathcal U$ on $\mathbb N$ we have $\mathcal{B}\left( H\right) '\cap \prod_{\mathcal U}\mathcal{B}\left( H\right) =
\mathbb C I$? \end{problem}




\end{problemblock}



\begin{problemblock}An ultrafilter $\mathcal U$ on $\mathbb N$ is \emph{flat} if there are $h_n\colon
\mathbb N\searrow [0,1]$ such that
\begin{enumerate}
\item $h_n(0)=1$,
\item $\lim_j h_n(j)=0$,
\item $(\forall f\colon \mathbb N\nearrow \mathbb N)\lim_{n\to \mathcal U}
\sup_{j\in \mathbb N} |h_n(j)-h_n(f(j))|=0$.
\end{enumerate}


\begin{problem}[.4] Is a nonprincipal ultrafilter such that $\mathcal{B}\left( H\right) '\cap \mathcal{B}\left( H\right) ^{\mathcal U}\neq \mathbb C I$ flat? \end{problem}




\end{problemblock}



\section{Calkin algebra}





\begin{problemblock}

\begin{problem}[.1] Is there a model of set theory in which the Calkin algebra has an
automorphism sending the unilateral shift $S$ to $S^{\ast }$? Is the
analogous fact for $\ell ^{\infty }\left/ c_{0}\right. $ true? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.2] Is there a model of set theory in which the Calkin algebra has a not
approximately inner automorphism preserving K-theory? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.3] Is it consistent that, for a nonseparable Hilbert space $\mathcal{H}$, 
$\mathcal{B}\left( \mathcal{H}\right) \left/ \mathcal{K}\left( \mathcal{H}%
\right) \right. $ has an outer automorphism? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.4] Does every masa in the Calkin algebra that is generated by its projections lift to a masa in $\mathcal{B}\left( H\right) $? \end{problem}




\end{problemblock}



\section{Tensor products}





\begin{problemblock}

\begin{problem}[.1] Does the number of C*-norms on $A\otimes _{alg}B$ for

\begin{itemize}
\item $A=B=\mathcal{B}\left( H\right) $

\item $A=B=C^{\ast }\left( \mathbb{F}_{\infty }\right) $, where $\mathbb{F}%
_{\infty }$ is the free group on countably many generators

\item $A=B\left( H\right) $ and $B=\mathcal{Q}\left( H\right) $ where $%
\mathcal{Q}\left( H\right) $ is the Calkin algebra
\end{itemize}

depend on the model of set theory? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.2] What is the ideal structure of $\mathcal{B}\left( H\right) \otimes
_{\min }\mathcal{Q}\left( H\right) $? Does it depend on the model of set
theory? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.3] Does $\otimes _{\min }$ or $\otimes _{\max }$ commute with forcing? \end{problem}




\end{problemblock}



\section{Nuclearity}





\begin{problemblock}

\begin{problem}[.1] Is every stably finite nuclear separable C*-algebra locally
approximable by type I C*-algebras? \end{problem}




\end{problemblock}



\begin{problemblock}All presently known nuclear C*-algebras are obtained from $\mathbb C$ by
taking closure under simple operations such as tensoring with
finite-dimensional matrix algebras and taking inductive limits. They
belong to the  so-called bootstrap class.

\begin{problem}[.2] Does every nuclear C*-algebra belong to the
bootstrap class? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.3] Do all nuclear C*-algebras satisfy the UCT (Universal Coefficient
Theorem)? \end{problem}




\end{problemblock}



\section{Pure states}





\begin{problemblock}

\begin{problem}[.1] Does every pure state of the atomic masa in $\mathcal{B}\left( H\right) $ extend uniquely to a pure
state of $\mathcal{B}\left( H\right) $? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.2] Consider the following statement: for every pure state $\phi$ of $\mathcal{B}\left( H\right) $ there is a masa $\mathcal{A}$ such that
$\phi\restriction \mathcal{A}$ is multiplicative (i.e., pure). Is it consistent with ZFC? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.3] Consider the following statement: every pure state on $\mathcal{B}\left( H\right) $ is diagonalizable. Is it consistent with with ZFC? \end{problem}




\end{problemblock}



\section{Nonseparable C*-algebras}





\begin{problemblock}

\begin{problem}[.1] Is there a nonseparable AF C*-algebra or W*-algebra not isomorphic to
its opposite algebra? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.2] Is the following relatively consistent with ZFC?

Every C*-algebra $A$ that has a unique irreducible representation up to the
unitary equivalence is isomorphic to the algebra of compact operators on some
Hilbert space. \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.3] Assume $A$ is a tensor product of algebras of the form $\mathbb{M}_n(\mathbb C)$, for $n\in \mathbb N$,
 $\kappa<\kappa'$ are cardinals
 and $\bigotimes_{\kappa'}\mathbb{M}_2(\mathbb C)$ unitally embeds into
$A\otimes \bigotimes_{\kappa}\mathbb{M}_2(\mathbb C)$. Can we conclude that there is a unital
embedding of $\bigotimes_\kappa \mathbb{M}_2(\mathbb C)$ into $A$? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.4] Is every exact C*-algebra a subalgebra of a nuclear C*-algebra? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.5] Is there a universal nuclear C*-algebra of character density $\aleph_1$? 
More generally, for which cardinals $\kappa$ is there a universal 
nuclear C*-algebra of character density $\kappa$? Similar question can be asked for exact algebras.  \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.6] Suppose a C*-algebra
has an approximate identity consisting of projections. Does it have an
increasing approximate identity (on some index set, possibly different)
consisting of projections? \end{problem}




\end{problemblock}



\section{Borel complexity}





\begin{problemblock}

\begin{problem}[.05] Is conjugacy by automorphism of masas of the hyperfinite II$_{1}$
factor classifiable by countable structures? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.1] What is the right set-theoretic framework to deal with functorial
classification? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.15] Is there a constructive Borel proof of the $\mathcal{O}_{2}$ embedding theorem? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.2] Is the isomorphism relation in the following classes of separable
C*-algebras:

\begin{itemize}
\item all

\item nuclear

\item exact

\item simple

\item simple exact

\item nuclear $\mathcal{Z}$-stable

\item simple nuclear
\end{itemize}

complete analytic? below a group actions? above all group actions? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.25] What is the complexity of isometric isomorphism of direct limits of
not necessarily self-adjoint subalgebras of finite dimensional C*-algebras?
How does it compare to the complexity of isomorphism of AF algebras? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.3] Is there a Borel inverse of the classification functor? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.35] For a separable C*-algebra $A$, what is the complexity of orbit
equivalence relations associated to the action $\mathrm{Aut}\left( A\right) $
on itself by conjugacy and to the action of $\mathrm{Inn}\left( A\right) $
on $\mathrm{\mathrm{Au}t}\left( A\right) $ by left translation? Are these
action turbulent? How are they related to structural properties of $A$? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.4] Is the Mackey Borel
structure on the spectrum of a simple separable C*-algebra always the same when
it is not standard? \end{problem}




\begin{remark}
\by{Elliott}All nonstandard spectra of AF algebras are isomorphic.\end{remark}

\end{problemblock}



\begin{problemblock}

\begin{problem}[.45] Does the complexity of the Mackey Borel structure of a simple separable
C*-algebra increase as one goes from nuclear C*-algebras to exact ones to ones
that are not even exact? \end{problem}




\end{problemblock}



\begin{problemblock}If A is a C*-algebra, denote by $E_A$ the relation of unitary equivalence of pure states of A.

\begin{problem}[.5]  Assume $A$ and $B$ are C*-algebras and $E_A$ is Borel-reducible to $E_B$. What does this fact imply about the
relation between $A$ and $B$? \end{problem}




\end{problemblock}



\section{Generators}





\begin{problemblock}A C*-algebra is called singly generated if it contains an element that is not contained in any proper sub-C*-algebra.
A C*-algebra is called $\mathcal{Z}$-stable if it absorbs the Jiang-Su algebra tensorially.


\begin{problem}[.1] Which separable, unital, simple C*-algebras are singly generated? Is there a simple, nuclear C*-algebra that is singly generated, yet is not $\mathcal{Z}$-stable? Is single generation connected to the regularity properties coming up in the classification of nuclear, simple C*-algebras? \end{problem}




\end{problemblock}





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\end{document}
