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Isometric Dilations of Non-Commuting Finite Rank n-Tuples

Published online by Cambridge University Press:  20 November 2018

Kenneth R. Davidson ,
David W. Kribs  and
Miron E. Shpigel
Kenneth R. Davidson
Affiliation:
Department of Pure Mathematics University of Waterloo Waterloo, Ontario N2L 3G1, e-mail: krdavids@uwaterloo.ca
David W. Kribs
Affiliation:
Department of Pure Mathematics University of Waterloo Waterloo, Ontario N2L 3G1, e-mail: dwkribs@uwaterloo.ca
Miron E. Shpigel
Affiliation:
Mitra Imaging Inc. 455 Phillip Street Waterloo, Ontario N2L 1W3, e-mail: mshpigel@mitra.com
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Abstract

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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 $B$.

Keywords

47L80
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 53 , Issue 3 , 01 June 2001 , pp. 506 - 545
Copyright
Copyright © Canadian Mathematical Society 2001

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