The ideal structure of the Haagerup tensor product of \(C^*\)-algebras.

*(English)*Zbl 0784.46040The Haagerup tensor \(A\otimes_ h B\) product of two \(C^*\)-algebras \(A\) and \(B\) is a Banach algebra with a natural contractive monomorphism from \(A\otimes_ h B\) into \(A\otimes_{\min} B\), where \(A\otimes_{\min} B\) denotes the minimal \(C^*\)-tensor product [D. Blecher, Math. Proc. Camb. Phil. Soc. 104, No. 1, 119-127 (1988; Zbl 0668.46027)]. Geometrically the Haagerup tensor product is well behaved and is injective. In this paper properties of closed ideals in the Haagerup tensor product \(A\otimes_ h B\) are studied. The minimal, maximal prime and primitive ideals are characterized in terms of the corresponding ideals in \(A\) and \(B\). For example, from this characterization the closed ideal lattice in \(B(H)\otimes_ h B(H)\) may be easily shown to contain four non-trivial closed ideals \(B(H)\otimes_ h K(H)+ K(H)\otimes_ h B(H)\), \(K(H)\otimes_ h B(H)\), \(B(H)\otimes_ h K(H)\) and \(K(H)\otimes_ h K(H)\) for a separable Hilbert space \(H\), where \(K(H)\) denotes the compact operators on \(H\).

The main tools are the algebraic and geometrical ideas mentioned above, the splitting lemma for a strongly independent sequence [Lemma 4.1, R. R. Smith, J. Funct. Anal. 102, No. 1, 156-175 (1991; Zbl 0745.46060)], and the following two lemmas concerning elementary tensors that are proved in the paper. There is an elementary tensor \(a\otimes b\) in each proper closed ideal in \(A\otimes_{\min} B\). If \(J\) is a closed ideal in \(A\otimes_ h B\), and \(a\otimes b\) is an elementary tensor in the closure if \(J\) in \(A\otimes_{\min} B\), then \(a\otimes b\) is already in \(J\).

The main tools are the algebraic and geometrical ideas mentioned above, the splitting lemma for a strongly independent sequence [Lemma 4.1, R. R. Smith, J. Funct. Anal. 102, No. 1, 156-175 (1991; Zbl 0745.46060)], and the following two lemmas concerning elementary tensors that are proved in the paper. There is an elementary tensor \(a\otimes b\) in each proper closed ideal in \(A\otimes_{\min} B\). If \(J\) is a closed ideal in \(A\otimes_ h B\), and \(a\otimes b\) is an elementary tensor in the closure if \(J\) in \(A\otimes_{\min} B\), then \(a\otimes b\) is already in \(J\).

Reviewer: S.D.Allen