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Orthogonal maximal abelian \(*\)-subalgebras of the \(n\times n\) matrices and cyclic \(n\)-roots. (English) Zbl 0914.46045

Doplicher, S. (ed.) et al., Operator algebras and quantum field theory. Proceedings of the conference dedicated to Daniel Kastler in celebration of his 70th birthday, Accademia Nazionale dei Lincei, Roma, Italy, July 1–6, 1996. Cambridge, MA: International Press. 296-322 (1997).
Two maximal Abelian \(*\)-subalgebras \(A\) and \(B\) of \(M_n(\mathbb{C})\), the algebra of complex \(n\times n\)-matrices, are said to be orthogonal if \(A\cap B= \mathbb{C} 1\) and if \(E_AE_B= E_{\mathbb{C} 1}\) for the corresponding orthogonal projections in \(M_n(\mathbb{C})\) (with respect to Hilbert-Schmidt norm). Up to a change of the orthonormal basis of \(\mathbb{C}^n\) the subalgebras \(A\) and \(B\) can be assumed to equal \(D\), the algebra of diagonal matrices and \(uDu^*\) resp., where \(u\) is some unitary matrix, and orthogonality of \(A\) and \(B\) is expressed by the condition \(| u_{jk}|= {1\over\sqrt n}\) for each \(j\) and \(k\). Examining the existence of orthogonal pairs of maximal Abelian \(*\)-subalgebras for different \(n\) it is shown that for \(n= 5\) all orthogonal pairs are isomorphic to the “standard” pair generated by the unitary matrix with entries \({1\over\sqrt n}\exp\left(i {2\pi\over n} jk\right)\). The investigation of unitary matrices operating on \(\mathbb{C}^n\) is embedded in the more general setting of determining cyclic \(n\)-roots, and this leads to further information for \(n= 6,7,8\) and for square free integers.
For the entire collection see [Zbl 0889.00022].
Reviewer: G.Garske (Hagen)

MSC:

46K05 General theory of topological algebras with involution
46H10 Ideals and subalgebras
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