Similarity and ergodic theory of positive linear maps

By Gelu Popescu. 2003

47 citations recorded by the source corpus.

In this paper we study the operator inequality \phi(X)\leq X and the operator equation \phi(X)= X, where \phi is a w^*-continuous positive (resp. completely positive) linear map on B(H). We show that their solutions are in one-to-one correspondence with a class of Poisson transforms on Cuntz-Toeplitz C^*-algebras, if \phi is completely positive. Canonical decompositions, ergodic type theorems, and lifting theorems are obtained and used to provide a complete description of all solutions, when \phi(I)\leq I. We show that the above-mentioned inequality (resp. equation) and the structure of its solutions have strong implications in connection with representations of Cuntz-Toeplitz C^*-algebras, common invariant subspaces for n-tuples of operators, similarity of positive linear maps, and numerical invariants associated with Hilbert modules over \CF_n^+, the complex free semigroup algebra generated by the free semigroup on n generators.

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

View Paper: https://www.semanticscholar.org/paper/be2dcadc3e8efb185287924aef5a6a6773c226eb

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Citations in the corpus

Free holomorphic functions and interpolation (2007, background): “In the last two decades, significant progress has been made in noncommutative multivariable operator theory regarding noncommutative dilation theory and its applications to interpolation in several variables ([24], [28], [40], [38], [30], [4], [36]), and unitary invariants for n-tuples of operators ”
Free holomorphic functions on the unit ball of B(H)n (2006, unknown): “...,Sn on the full Fock space with ngenerators, F2(Hn), and the identity. These algebras have been intensively studied in recent years by many authors ([27], [28], [29], [30], [31], [32], [34], [35], [36], [2], [3], [16], [17], [13], [24], [40]). The results of this section are used to obtain a maxi”
Unitary Invariants in Multivariable Operator Theory (2004, background): “Refinements of these unitary invariants were considered in [ 56 ].”
Operator Theory on Noncommutative Domains (2007, unknown): “Introduced in [70] in connection with a multivariable noncommutative von Neumann inequality, the noncommutative analytic Toeplitz algebra F∞ n has been studied in several papers [68], [69], [72], [73], [74], [76], [6], and recently in [35], [36], [37], [8], [78], [39], [64], [81], and [87].”
Operator theory on noncommutative varieties (2005, background): “More about noncommutative Poisson transforms on C∗-algebras generated by isometries can be found in [30], [1], [31], [32], and [34].”
Entropy and Multivariable Interpolation (2003, background): “The noncommutative analytic Toeplitz algebra F ∞ n has been studied in several papers [39], [40], [43], [44], [45], [47], [2], and recently in [14], [15], [16], [4], [51], [13], [36], and [56].”