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# Mathematical Research Letters

## Volume 6 (1999)

### Number 2

### Convex sets associated with von~Neumann algebras and Connes' approximate embedding problem

Pages: 229 – 236

DOI: http://dx.doi.org/10.4310/MRL.1999.v6.n2.a11

#### Author

#### Abstract

Connes' approximate embedding problem, asks whether any countably generated type ${II}_1$ factor $M$ can be approximately embedded in the hyperfinite type ${II}_1$ factor. Solving this problem in the affirmative, amounts to showing that given any integers $N,p$, any elements $x_1,\dots,x_N$ in $M$ and any $\epsilon >0$, one can find $k$ and matrices $X_1, \dots, X_N$ in the algebra $M_k(\Bbb C)$, endowed with the normalized trace $\text{\ tr}$, such that for every $i_1,\dots,i_p\in \{1,\dots,N\}$ and for every $s$, with $1\leq s\leq p$, one has that $|\tau (x_{i_1}\dots x_{i_s})-\text{\ tr\ }(X_{i_1}\dots X_{i_s})|<\epsilon.$ In this paper we show that this is always possible if $s$ is 2 and 3. More precisely we prove that for every strictly positive integer $N$, for every elements $x_1,\dots,x_N$ in $M$ and any $\epsilon >0$, one can find $k$ and matrices $X_1, \dots, X_N$ in the algebra $M_k(\Bbb C)$, such that for every $i_1,i_2,i_3\in \{1,\dots,N\}$ one has that $|\tau (x_{i_1}x_{i_2}x_{i_3})-\text{\ tr}(X_{i_1}X_{i_2}X_{i_3})|<\epsilon$ and $|\tau (x_{i_1}x_{i_2})-\text{\ tr\ }(X_{i_1}X_{i_2})|<\epsilon.$ An affirmative solution of the Connes' problem would follow if the above statement could also be proved for $s=4$.