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Completely Bounded Maps and Operator Algebras

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Vern Paulsen at University of Waterloo
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... The first inequality in (2.9) follows from [30, Proposition 2.1.1] and [28,Proposition 3.6]. Hence, we get (2.8) and it follows that D Q is a matrix convex nc set. ...
... Since every state is also UCP (see [28,Proposition 3.8]), we get that φ v (X) ∈ D Q (1). However, by (2.11), we get that ...
Weak-* and completely isometric structure of noncommutative function algebras
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We study operator algebraic and function theoretic aspects of algebras of bounded nc functions on subvarieties of the nc domain determined by all levels of the unit ball of an operator space (nc operator balls). Our main result is the following classification theorem: under very mild assumptions on the varieties, two such algebras H∞(V)H^\infty(\mathfrak{V}) and H∞(W)H^\infty(\mathfrak{W}) are completely isometrically and weak-* isomorphic if and only if there is a nc biholomorphism between the varieties. For matrix spanning homogeneous varieties in injective operator balls, we further sharpen this equivalence, showing that there exists a linear isomorphism between the respective balls that maps one variety onto the other; in general, we show, the homogeneity condition cannot be dropped. We highlight some difficulties and open problems, contrasting with the well studied case of row ball.
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... 2 This is the completely bounded norm of T when regarded as an element of ℓ 1 n ⊗ h · · · ⊗ h ℓ 1 n , where h stands for the Haagerup tensor product, which determines a well-studied tensor norm. See for instance [Pau03, Chapter 17]. ...
Grothendieck inequalities characterize converses to the polynomial method
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A surprising 'converse to the polynomial method' of Aaronson et al. (CCC'16) shows that any bounded quadratic polynomial can be computed exactly in expectation by a 1-query algorithm up to a universal multiplicative factor related to the famous Grothendieck constant. Here we show that such a result does not generalize to quartic polynomials and 2-query algorithms, even when we allow for additive approximations. We also show that the additive approximation implied by their result is tight for bounded bilinear forms, which gives a new characterization of the Grothendieck constant in terms of 1-query quantum algorithms. Along the way we provide reformulations of the completely bounded norm of a form, and its dual norm.
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... All constants are computed explicitly. We note that the result can be extended to the case of rectangular matrices using dilations (see, e.g., [Paulsen, 2002]). ...
Bernstein-type and Bennett-type inequalities for unbounded matrix martingales
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  • Nov 2024
We derive explicit Bernstein-type and Bennett-type concentration inequalities for matrix-valued supermartingale processes with unbounded observations. Specifically, we assume that the ψα\psi_{\alpha}-Orlicz (quasi-)norms of their difference process are bounded for some α>0\alpha > 0. As corollaries, we prove an empirical version of Bernstein's inequality and an extension of the bounded differences inequality, also known as McDiarmid's inequality.
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... We start by briefly recalling the theory of operator systems, referring to [8,17,18,3] for more details. ...
Higher K-groups for operator systems
Preprint
Full-text available
  • Nov 2024
We extend our previous definition of K-theoretic invariants for operator systems based on hermitian forms to higher K-theoretical invariants. We realize the need for a positive parameter δ\delta as a measure for the spectral gap of the representatives for the K-theory classes. For each δ\delta and integer p≥0p \geq 0 this gives operator system invariants Vpδ(−,n)\mathcal V_p^\delta(-,n), indexed by the corresponding matrix size. The corresponding direct system of these invariants has a direct limit that possesses a semigroup structure, and we define the KpδK_p^\delta-groups as the corresponding Grothendieck groups. This is an invariant of unital operator systems, and, more generally, an invariant up to Morita equivalence of operator systems. Moreover, there is a formal periodicity that reduces all these groups to either K0δK_0^\delta or K1δK_1^\delta. We illustrate our invariants by means of the spectral localizer.
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On von Neumann’s inequality on the polydisc
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  • MATH ANN
Given a d-tuple T of commuting contractions on Hilbert space and a polynomial p in d-variables, we seek upper bounds for the norm of the operator p(T). Results of von Neumann and Andô show that if d=1 or d=2, the upper bound ∥p(T)∥≤∥p∥∞\Vert p(T)\Vert \le \Vert p\Vert _\infty , holds, where the supremum norm is taken over the polydisc Dd\mathbb {D}^d. We show that for d=3, there exists a universal constant C such that ∥p(T)∥≤C∥p∥∞\Vert p(T)\Vert \le C \Vert p\Vert _\infty for every homogeneous polynomial p. We also show that for general d and arbitrary polynomials, the norm ∥p(T)∥\Vert p(T)\Vert is dominated by a certain Besov-type norm of p.
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Matrix convexity and unitary power dilations of Toeplitz-contractive operator tuples
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Using works of T. Ando and L. Gurvits, the well-known theorem of P.R. Halmos concerning the existence of unitary dilations for contractive linear operators acting on Hilbert spaces is recast as a result for d -tuples of contractive Hilbert space operators satisfying a certain matrix-positivity condition. Such operator d -tuples satisfying this matrix-positivity condition are called, herein, Toeplitz-contractive, and a characterisation of the Toeplitz-contractivity condition is presented. The matrix-positivity condition leads to definitions of new distance-measures in several variable operator theory, generalising the notions of norm, numerical radius, and spectral radius to d -tuples of operators (commuting, for the spectral radius) in what appears to be a novel, asymmetric way. Toeplitz contractive operators form a noncommutative convex set, and a scaling constant cdc_d c d for inclusions of the minimal and maximal matrix convex sets determined by a stretching of the unit circle S1S^1 S 1 across d complex dimensions is shown to exist.
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The 2×22 \times 2 block matrices associated with an annulus
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A bounded Hilbert space operator T for which the closure of the annulus is a spectral set is called an Ar\mathbb {A}_r-contraction. A celebrated theorem due to Douglas, Muhly, and Pearcy gives a necessary and sufficient condition such that a 2×22 \times 2 block matrix of operators [T1X0T2] \begin{bmatrix} T_1 & X \\ 0 & T_2 \end{bmatrix} is a contraction. We seek an answer to the same question in the setting of an annulus, i.e., under what conditions does T~Y=[T1Y0T2]\widetilde{T}_Y=\begin{bmatrix} T_1 & Y\\ 0 & T_2\\ \end{bmatrix} become an Ar\mathbb {A}_r-contraction? For Ar\mathbb {A}_r-contractions T,T1,T2T, T_1,T_2 and an operator X that commutes with T,T1,T2T, T_1,T_2, here we find a necessary and sufficient condition such that each of the block matrices becomes an Ar\mathbb {A}_r-contraction.
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Operator product states on tensor powers of C∗C^*-algebras
Article
  • Oct 2024
The program of matrix product states on tensor powers A⊗Z{\mathcal {A}}^{\otimes {\mathbb {Z}}} of C∗C^*-algebras is carried under the assumption that A{\mathcal {A}} is an arbitrary nuclear C*-algebra. For any shift invariant state ω\omega , we demonstrate the existence of an order kernel ideal Kω{\mathcal {K}}_\omega , whose quotient action reduces and factorizes the initial data (A⊗Z,ω)({\mathcal {A}}^{\otimes {\mathbb {Z}}}, \omega ) to the tuple (A,Bω=A⊗N×/Kω,Eω:A˚⊗Bω→Bω,ωˉ:Bω→C)({\mathcal {A}},{\mathcal {B}}_\omega = {\mathcal {A}}^{\otimes {\mathbb {N}}^\times }/{\mathcal {K}}_\omega , {\mathbb {E}}_\omega : \text{\AA }\otimes {\mathcal {B}}_\omega \rightarrow {\mathcal {B}}_\omega , {\bar{\omega }}: {\mathcal {B}}_\omega \rightarrow {\mathbb {C}}), where Bω{\mathcal {B}}_\omega is an operator system and Eω{\mathbb {E}}_\omega and ωˉ{\bar{\omega }} are unital and completely positive maps. Reciprocally, given a (input) tuple (A,S,E,ϕ)({\mathcal {A}},{\mathcal {S}},{\mathbb {E}},\phi ) that shares similar attributes, we supply an algorithm that produces a shift-invariant state on A⊗Z{\mathcal {A}}^{\otimes {\mathbb {Z}}}. We give sufficient conditions in which the so constructed states are ergodic and they reduce back to their input data. As examples, we formulate the input data that produces AKLT-type states, this time in the context of infinite dimensional site algebras A{\mathcal {A}}, such as the C∗C^*-algebras of discrete amenable groups.
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Choi representation of completely positive maps in brief
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The Choi representation of completely positive trace preserving (CPTP) maps, i.e. quantum channels is often used in the context of quantum information and computation as it is easy to work with. It is a correspondence between CPTP maps and quantum states also termed as the Choi–Jamiołkowski isomorphism. It is especially useful if a parametrization of the set of CPTP maps is needed in order to consider a general map or optimize over the set of these. Here we provide a brief introduction to this topic, focusing on certain useful calculational techniques which are presented in full detail.
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