## 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)].

### 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

Zbl 0554.46026
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