Isometric Dilations of Non-Commuting Finite Rank n-Tuples

By K. Davidson, David W. Kribs, Miron E. Shpigel. Canadian Journal of Mathematics - Journal Canadien de Mathematiques, 2001

66 citations recorded by the source corpus.

Abstract 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 $B$ .

Subfields: Algebra and Number Theory, Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9b50ca310f42cbb1742c3b485323a81ede4a7fc1

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Citations in the corpus

Dilation theory, commutant lifting and semicrossed products (2011, background): “By [20], the wandering space W of N is given by”
Similarity and ergodic theory of positive linear maps (2003, background): “We will refer to these sets as the C(ϕ)-sets associated with ϕ. The structure of these sets plays a distinguished role in the ergodic theory of positive maps [21], the classification of the endomorphisms of B(H) (eg.[41], [42], [5], [6], [25], [11], [12], [13]), and the representation theory of Cunt”
Operator theory on noncommutative varieties (2005, result): “In the noncommutative case, significant results were obtained in [14], [9], [20], [21], [22], [24], and recently in [ 10 ].”
Free holomorphic functions on the unit ball of B(H)n (2006, unknown): “2 GELU POPESCU In recent years, there has been exciting progress in noncommutative multivariable operator theory regarding noncommutative dilation theory ([19], [10], [25], [26], [27], [14], [8], [9], [40], [39], etc.”
The isomorphism problem for some universal operator algebras (2010, background): “, Vd} may fall into several quite different isomorphism classes: it might be Ld , it might be a type I∞ factor, and it might be something “in between” (see [15,16,19,31]).”
Noncommutative Choquet Theory (2019, unknown): “In [18], it is shown that a finite dimensional coisometric tuple dilates to a representation of Od having the special property that there is a finite dimensional coinvariant cyclic invariant subspace for the d generators.”
Operator Theory on Noncommutative Domains (2007, unknown): “-Nagy–Foiaş theory (see [45], [29], [66], [67], [68], [69], [70], [71], [72], [73], [74] and more recently [76], [38], [17], [18], [23], [77], [78], [80], [82], [86], [83], [84], [85], [21]).”