Conversion from nonstandard matrix algebras to standard factors of type \(II_ 1\)
type: Item
instance of: scholarly article
title: Conversion from nonstandard matrix algebras to standard factors of type \(II_ 1\)
zbMATH Open document ID: 0803.03047
author: Takanori Hinokuma
author: Masanao Ozawa
published in: Illinois Journal of Mathematics
publication date: 1995-01-03T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 03H05
Mathematics Subject Classification ID: 46L35
Mathematics Subject Classification ID: 46S20
zbMATH DE Number: 561187
zbMATH Keywords: nonstandard matrix algebras
zbMATH Keywords: standard factors
zbMATH Keywords: complex matrices
zbMATH Keywords: nonstandard hull
zbMATH Keywords: quotient space
zbMATH Keywords: non-separable Hilbert space
zbMATH Keywords: approximately finite
reviewed by: W. A. J. Luxemburg
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 561187