On the \(K_ 0\) groups of nest algebras
The \(K_ 0\) functor is computed on a class of subalgebras of \({\mathcal B}({\mathcal H})\), the so called nest subalgebras. Such algebras \(A(\beta)\) are associated to given nests \(\beta\) of projections in \(B({\mathcal H})\). Define a linear mapping \(\phi\) of \({\mathcal B}({\mathcal H})\) by \(\phi (X)=0\) if \(\beta\) is continuous, and \(\sum q_ iXq_ i\) otherwise where \(\{q_ i\}\) is the set of atoms of \(\beta\). Let \(Q=\phi (I)\) and \(D(\beta)=A(\beta)\cap A(\beta)^*\). The author proves that the homomorphism \(\phi: A(\beta)\to QD(\beta)Q\) induces an isomorphism, \(\phi_*: K_ 0(A(\beta))\to K_ 0(QD(\beta)Q)\). If \(\Lambda\) denotes the set of finite-dimensional atoms in \(\beta\), then the author proves that \(K_ 0(A(\beta))\simeq {\bar \oplus}_{q\in \Lambda}{\mathbb{Z}}\) where \({\bar \oplus}_{q\in \Lambda}{\mathbb{Z}}=\{g: \Lambda \to {\mathbb{Z}}:\quad \sup_{\Lambda}| g(q)/tr(q)| <\infty \}.\)
scientific article; zbMATH DE number 4117247
scientific article