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**Diagonalizing matrices over \(AW^\ast\)-algebras.**
*(English)*
Zbl 1279.15015

Let \(M_n({\mathcal A})\) be the matrix algebra of \(n\times n\) matrices with coefficients from a unital \(AW^\ast\) algebra \({\mathcal A}\). The main result shows that for each integer \(n\geq 2\) every commuting subset \(X\subseteq M_n({\mathcal A})\) of normal matrices is simultaneously diagonalizable in a sense that there exists a unitary \(u\in M_n({\mathcal A})\) such that \(uxu^\ast\) are diagonal matrices for all \(x\in X\). For a proof, the authors study the dimension function, \(d(e)\), of a properly infinite projection \(e\), which is defined to be the least cardinal strictly larger than the cardinal of any orthogonal family of projections equivalent to \(e\). It is shown, for example, that for two properly infinite projections \(e,f\) which have the same central cover, \(e\sim f'\leq f\) implies \(d(e)\leq d(f)\). This may no longer be true if their central covers, \(c(e)\) and \(c(f)\), differ. However, if \(e,f\) belong to a certain large class of properly infinite projections, then \(e\sim f'\leq f\) is equivalent to the fact that \(d(e)\leq d(f)\) and \(c(e)\leq c(f)\). This class will not be defined in the present review; let us just mention that every properly infinite projection in \({\mathcal A}\) is a sum of orthogonal projections from the class.

As an application it is shown that if \(\phi:{\mathcal A}\to{\mathcal B}\) is a \(\ast\)-homomorphism between unital \(AW^\ast\)-algebras, which preserves the suprema of subsets of projections, then the same is true for a map \(M_n(\phi):M_n({\mathcal A})\to M_n({\mathcal B})\), defined by \((a_{ij} )_{ij}\mapsto (\phi(a_{ij}) )_{ij}\). It follows that the assignment \(\mathbb M_n:{\mathcal A}\mapsto M_n({\mathcal A})\), which transforms the morphism \({\mathcal A}\buildrel\phi\over\longrightarrow {\mathcal B}\) into \(M_n({\mathcal A})\buildrel M_n(\phi)\over\longrightarrow M_n({\mathcal B})\), is a functor in the category AWstar whose objects are unital \(AW^\ast\)-algebras and whose morphisms are \(\ast\)-homomorphisms that preserve the suprema of subsets of projections.

The converse implication, that is, if a unital \(C^\ast\)-algebra \({\mathcal C}\) must be an \(AW^\ast\)-algebra when every commuting subset \(X\subseteq M_n({\mathcal C})\) of normal matrices is, for each integer \(n\geq 2\), simultaneously diagonalizable, remains unknown at the time of writing the report. It is known, however, to be true for abelian \(C^\ast\)-algebras by K. Grove and G. K. Pedersen [J. Funct. Anal 59, 65–89 (1984; Zbl 0554.46026)].

As an application it is shown that if \(\phi:{\mathcal A}\to{\mathcal B}\) is a \(\ast\)-homomorphism between unital \(AW^\ast\)-algebras, which preserves the suprema of subsets of projections, then the same is true for a map \(M_n(\phi):M_n({\mathcal A})\to M_n({\mathcal B})\), defined by \((a_{ij} )_{ij}\mapsto (\phi(a_{ij}) )_{ij}\). It follows that the assignment \(\mathbb M_n:{\mathcal A}\mapsto M_n({\mathcal A})\), which transforms the morphism \({\mathcal A}\buildrel\phi\over\longrightarrow {\mathcal B}\) into \(M_n({\mathcal A})\buildrel M_n(\phi)\over\longrightarrow M_n({\mathcal B})\), is a functor in the category AWstar whose objects are unital \(AW^\ast\)-algebras and whose morphisms are \(\ast\)-homomorphisms that preserve the suprema of subsets of projections.

The converse implication, that is, if a unital \(C^\ast\)-algebra \({\mathcal C}\) must be an \(AW^\ast\)-algebra when every commuting subset \(X\subseteq M_n({\mathcal C})\) of normal matrices is, for each integer \(n\geq 2\), simultaneously diagonalizable, remains unknown at the time of writing the report. It is known, however, to be true for abelian \(C^\ast\)-algebras by K. Grove and G. K. Pedersen [J. Funct. Anal 59, 65–89 (1984; Zbl 0554.46026)].

Reviewer: Bojan Kuzma (Ljubljana)

### MSC:

15A30 | Algebraic systems of matrices |

46L05 | General theory of \(C^*\)-algebras |

46L10 | General theory of von Neumann algebras |

15A21 | Canonical forms, reductions, classification |

15A86 | Linear preserver problems |