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\(K\)-Groups of a \(C^{*}\)-Algebra Generated by a Single Operator

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Abstract

In this paper, we compute \(K\)-groups \(\{K_{n}(C^{*}(x))\}_{n=0}^{\infty }\) of the \(C^{*}\)-subalgebra \(C^{*}(x)\) of \(B(H),\) generated by a single operator \(x,\) where \(H\) is a separable infinite dimensional Hilbert space, and \(B(H)\) is the operator algebra consisting of all (bounded linear) operators on \(H.\) These computations not only provide nice examples in \(K\)-theory, but also characterize-and-classify projections in a \(C^{*}\)-algebra generated by a single operator. The main result of this paper shows that: the \(K\)-groups of \(C^{*}(x)\) are completely characterized by those of \(C^{*}(q),\) where \(q\) is the positive-operator part of \(x\) in the polar decomposition of \(x.\)

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  1. Department of Mathematics, St. Ambrose University, 518 W. Locust St., Davenport, IA, 52803, USA

    Ilwoo Cho

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  1. Ilwoo Cho
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Correspondence to Ilwoo Cho.

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Communicated by Palle Jorgensen.

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Cho, I. \(K\)-Groups of a \(C^{*}\)-Algebra Generated by a Single Operator. Complex Anal. Oper. Theory 8, 1405–1434 (2014). https://doi.org/10.1007/s11785-013-0285-7

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