Isometric dilations of non-commuting finite rank $n$-tuples

Details Isometric dilations of non-commuting finite rank $n$-tuples Isometric dilations of non-commuting finite rank $n$-tuples

2004-11-23

arxiv.org/abs/math/0411521v1

Can. J. Math. 53 (2001), 506-545

Mathematics

Operator Algebras

46 pages, preprint version

Scientific paper

A contractive $n$-tuple $A=(A_1,...,A_n)$ has a minimal joint isometric dilation $S=(S_1,...,S_n)$ where the $S_i$'s are isometries with pairwise orthogonal ranges. This determines a representation of the Cuntz-Toeplitz algebra. When $A$ acts on a finite dimensional space, the \wot-closed nonself-adjoint algebra $\mathfrak{S}$ generated by $S$ is completely described in terms of the properties of $A$. This provides complete unitary invariants for the corresponding representations. In addition, we show that the algebra $\mathfrak{S}$ is always hyper-reflexive. In the last section, we describe similarity invariants. In particular, an $n$-tuple $B$ of $d\times d$ matrices is similar to an irreducible $n$-tuple $A$ if and only if a certain finite set of polynomials vanish on $B$.

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