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# Isometric Dilations of Non-Commuting Finite Rank $n$-Tuples

Published:2001-06-01
Printed: Jun 2001
• Kenneth R. Davidson
• David W. Kribs
• Miron E. Shpigel
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## Abstract

A contractive $n$-tuple $A=(A_1,\dots,A_n)$ has a minimal joint isometric dilation $S=\break (S_1,\dots,S_n)$ 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 $\fS$ 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 $\fS$ 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 $B$.
 MSC Classifications: 47L80 - Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.)