Hilbert \(C^*\)-bimodules and continuous Cuntz-Krieger algebras. (English) Zbl 1031.46067

Kajiwara, Tsuyoshi; Watatani, Yasuo

J. Math. Soc. Japan 54, No. 1, 35-59 (2002).

The authors consider certain correspondences on the disjoint union \(\Omega\) of circles which naturally give rise to Hilbert \(C^*\)-bimodules \(X\) over circle algebras \(A\). The bimodules \(X\) generate \(C^*\)-algebras \({\mathcal O}_X\) which are isomorphic to continuous Cuntz-Krieger algebras considered by { it V. Deaconu} [see Proc. Am. Math. Soc. 124, 3427-3435 (1996; Zbl 0864.46033)]. The simplicity and the ideal structure of the algebras \({\mathcal O}_X\) under some conditions using \((I)\)-freeness and \((II)\)-freeness defined in [T. Kajiwara, C. Pinzari and Y. Watatami, J. Funct. Anal. 159, 295-322 (1998; Zbl 0942.46035)] are studied. More precisely, a bijective correspondence between the set of closed two sided ideals of \({\mathcal O}_X\) and saturated hereditarily open subsets of \(\Omega\) is obtained.

Reviewer: L.Maxim-Răileanu (Iaşi)

MSC:

46L08	\(C^*\)-modules
19K99	\(K\)-theory and operator algebras
46L05	General theory of \(C^*\)-algebras
46L80	\(K\)-theory and operator algebras (including cyclic theory)

Keywords:

Hilbert \(C^*\)-bimodules

Citations:

Zbl 0864.46033; Zbl 0942.46035