Note on property \(({\mathbb{A}}_ 1)\). (English) Zbl 0622.15005
It is shown that there is a subspace N of \({\mathbb{C}}^{3\times 3}\) such that the bilinear map \(x\otimes y\) from \({\mathbb{C}}^ 3\times {\mathbb{C}}^ 3\) to \({\mathbb{C}}^{3\times 3}\) mod N is a (continuous) surjection but not open at the origin. With \({\mathbb{C}}^{7\times 7}\) instead of \({\mathbb{C}}^{3\times 3}\), N can be chosen such that the annihilator of N, with respect to the trace pairing, is a communicative algebra.
Reviewer: J.de Graaf
MSC:
| 15A30 | Algebraic systems of matrices |
| 46J05 | General theory of commutative topological algebras |
| 16P10 | Finite rings and finite-dimensional associative algebras |
| 47B10 | Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.) |
Keywords:
algebra of bounded linear operators; Hilbert space; trace-class operators; communicative algebraReferences:
| [1] | Bercovici, H.; Foiaş, C.; Pearcy, C., Dual Algebras with Applications to Invariant Subspaces and Dilation Theory, CBMS Conf. Ser. in Math., No. 56 (1985), Amer. Math. Soc: Amer. Math. Soc Providence · Zbl 0608.47005 |
| [2] | B. Chevreau, and C. Esterle, private communication.; B. Chevreau, and C. Esterle, private communication. |
| [3] | Cohen, P., A counterexample to the closed graph theorem for bilinear maps, J. Funct. Anal., 16, 235-239 (1974) · Zbl 0279.46002 |
| [4] | P. Dixon, private communication.; P. Dixon, private communication. |
| [5] | Hadwin, D.; Nordgren, E., Subalgebras of reflexive algebras, J. Operator Theory, 7, 3-23 (1982) · Zbl 0483.47023 |
| [6] | Horowitz, C., An elementary counterexample to the open mapping principle for bilinear maps, Proc. Amer. Math. Soc., 53, 293-294 (1975) · Zbl 0324.15015 |
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