Conversion from nonstandard matrix algebras to standard factors of type \(II_ 1\)
Let \(\nu\) be a nonstandard natural number in an \(\aleph\)-saturated nonstandard universe, and let \(M:= {^*M}(\nu)\) be the internal algebra of all the \(\nu\times \nu\)-complex matrices. \(p_ \infty\) denotes the usual operator norm on \(M\) with respect to the classical inner product norm on \({^*\mathbb{C}}^ \nu\). \(\tau\) denotes the usual normalized trace on \(M\),\((x| y):= \tau(y^* x)\quad (x,y\in M)\) is the inner product on \(M\) generated by \(\tau\), and the corresponding Hilbert-Schmidt norm on \(M\) is denoted by \(p_ 2\). \(\widehat M_ 2\) denotes the nonstandard hull of \((M,p_ 2)\). It is easy to see that \(\widehat M_ \infty\) is a \(C^*\)-algebra. The authors denote by \(\widehat M\) the quotient space \(\text{fin}_ \infty(M)/(\mu_ 2(0)\cap\text{fin}_ \infty(M))\). It is shown that \(\widehat M\) is a factor of \(\text{II}_ 1\). Furthermore, it is shown that every non-trivial representation of \(\widehat M\) must have a non-separable Hilbert space. The authors also observe that \(\widehat M\) is not approximately finite. The authors pose the problem whether any two \(\widehat M(\nu)\) and \(\widehat M(\mu)\) are isomorphic. They show that the answer is yes if \(\nu-\mu\) is finite.
scientific article; zbMATH DE number 561187