\(C^*\)-algebras generated by elements of a unitary matrix. (English) Zbl 0777.46033
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 \(U_ n^{nc}\).
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 \(U_ n^{nc}\).
Reviewer: M.Fritzsche (Potsdam)
MSC:
| 46L05 | General theory of \(C^*\)-algebras |
| 46L80 | \(K\)-theory and operator algebras (including cyclic theory) |
| 19K14 | \(K_0\) as an ordered group, traces |
Keywords:
unital \(C^*\)-algebra; unital \(*\)-homomorphism; amalgamated free product; nontrivial projectionsReferences:
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