The \(C^*\)-algebra generated by strong \(\Theta\)-operators and its K- theory (Q909947)

A bounded operator T in a separable Hilbert space is called a \(\Theta\)- operator, write \(T\in \Theta\), iff \(T^*[T^*,T]=[T^*,T]T.\) If in addition \(T^*_{| \overline{Ran(D_ T)}}\) is M-hyponormal, it will be called a strong \(\Theta\)-operator, write \(T\in S\Theta\). In the last case T is unitarily equivalent to the ``block-diagonal'' matrix \(\hat A+V\hat B\) (see Proposition 2). It is shown in the paper under reviewing that the commutant ideal I(T) of the \(C^*\)-algebra \(C^*(T)\) generated by T is *-isomorphic to the closure \(\overline{\cup_{n}Mat(C_ 0(Z_ 0))};\) \(Z_ 0=\{\lambda +i\mu \in \sigma (A+iB);\quad \mu \neq 0\}.\) If T is a completely non- normal strong \(\Theta\)-operator, I(T) is *-isomorphic to \(C_ 0(Z_ 0)\otimes K(\ell^ 2)\). For any completely nonnormal strong \(\Theta\)- operator T, \(C^*(T)\) is stable, i.e. each isometry in \(M_ n(C^*(T))\) is unitary for any n, if and only if the spectrum of T can be identified with the approximative spectrum. This statement is equivalent also to the vanishing of the \(K^{-1}\)-group; \(K^{- 1}(\sigma_{ap}(T))=0\). Some other K-groups are also computed.