\(C^*\)-algebras generated by elements of a unitary matrix
Let \(U_ n^{nc}\) be the unital \(C^*\)-algebra with the following universal property: There are elements \(u_{ij}\in U_ n^{nc}\) (\(1\leq i,j\leq n\)) such that the matrix \([u_{ij}]\), is a unitary in \(M_ n(U_ n^{nc})\) and if another unital \(C^*\)-algebra \(A\) contains elements \(v_{ij}\) with the same property, then there is a unique unital \(*\)-homomorphism \(\varphi: U+n^{nc}\to A\) such that \(\varphi(u_{ij})=v_{ij}\) (\(i,j=1,\dots,n\)). It is shown that \(U_ n^{nc}\) is isomorphic to the relative commutant \(M_ n^ c\) of \(M_ n\) in the amalgamated free product \(M_ n*_ \mathbb{C} C(T)\) (\(T\) denotes the unit circle) and that \(U_ n^{nc}\) has no nontrivial projections. Using the result \(K_ j(M_ n*_ \mathbb{C} A)\cong K_ j(A)\) (\(n\geq 1\), \(j=0,1\)) for a unital \(C^*\)-algebra \(A\) admitting a unital \(*\)- homomorphism \(w: A\to \mathbb{C}\), the author proves \(K_ j(U_ n^{nc})\cong\mathbb{Z}\) \((j=0,1)\). Further, a reduced version \(U^{nc}_{n,red}\) of \(U_ n^{nc}\) is defined and it is shown that \(U^{nc}_{n,red}\) is simple, has no nontrivial projections and is a homomorphic image of \(U_ n^{nc}\).
scientific article; zbMATH DE number 65350
scientific article