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The ideal structure of the Haagerup tensor product of $$C^*$$-algebras. (English) Zbl 0784.46040
The 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 $$J$$.
Reviewer: S.D.Allen

##### MSC:
 46L05 General theory of $$C^*$$-algebras 46M05 Tensor products in functional analysis
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