Noncommutative reproducing kernel Hilbert spaces

Ball, Joseph A.; Marx, Gregory; Vinnikov, Victor

doi:10.1016/j.jfa.2016.06.010

Cited by 51 publications

(72 citation statements)

References 44 publications

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“…If one replaces the commutative free Nullstellensatz with the noncommutative homogeneous Nullstellensatz, then the same argument as above (where polynomials are replaced by free polynomials) gives the following theorem: 6 Theorem 2.2. Let I and J be two homogeneous ideals in C z .…”

Section: A Digression -The Purely Algebraic Casementioning

confidence: 99%

Algebras of noncommutative functions on subvarieties of the noncommutative ball: The bounded and completely bounded isomorphism problem

Salomon

Shalit

Shamovich

2020

Journal of Functional Analysis

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Given a noncommutative (nc) variety V in the nc unit ball B d , we consider the algebra H ∞ (V) of bounded nc holomorphic functions on V. We investigate the problem of when two algebras H ∞ (V) and H ∞ (W) are isomorphic. We prove that these algebras are weak- * continuously isomorphic if and only if there is an nc biholomorphism G : W → V between the similarity envelopes that is bi-Lipschitz with respect to the free pseudohyperbolic metric. Moreover, such an isomorphism always has the form f → f • G, where G is an nc biholomorphism. These results also shed some new light on automorphisms of the noncommutative analytic Toeplitz algebras H ∞ (B d ) studied by Davidson-Pitts and by Popescu. In particular, we find that Aut(H ∞ (B d )) is a proper subgroup of Aut( B d ).When d < ∞ and the varieties are homogeneous, we remove the weak- * continuity assumption, showing that two such algebras are boundedly isomorphic if and only if there is a bi-Lipschitz nc biholomorphism between the similarity envelopes of the nc varieties. We provide two proofs. In the noncommutative setting, our main tool is the noncommutative spectral radius, about which we prove several new results. In the free commutative case, we use a new free commutative Nullstellensatz that allows us to bootstrap techniques from the fully commutative case. 1 arXiv:1806.00410v4 [math.OA] 4 Apr 2019 functions, in analogy with Kaplanski's theorem [25, Theorem 2] that states that elements of the free associative algebra C z 1 , . . . , z d are determined by their values on M d .Not surprisingly, however, M d is in a sense too big to have a rich theory of holomorphic functions, so just like in the classical case, analysts usually consider only certain subsets of it. Every classical domain, such as a ball or a polydisc admits natural "quantizations". In particular, in this paper we will focus on the nc ball B d : the set of all d-tuples (X 1 , . . .It is worth noting that the for every n, the nth level of the nc universe admins a natural GL n -action given by S · (X 1 , . . . , X d ) = (S −1 X 1 S, . . . , S −1 X d S). Unfortunately, most domains, including our nc ball, are not invariant under this action. We therefore define, for every set Ω ⊆ M d , its similarity orbit Ω, which is just the orbit of Ω under the levelwise GL n -action.The algebra of bounded nc holomorphic functions on the nc ball, H ∞ (B d ) turns out to be the free semigroup algebra L d studied by Arias and Popescu and Davidson and Pitts, see for example [3,11,12,30,34,38]. The free semigroup algebra is the universal weak- * closed algebra generated by a pure row contraction. Its quotients by weak- * closed two sided ideals are thus universal weak- * closed algebras generated by pure row contractions satisfying prescribed algebraic relations. We would like to understand when such algebras are isomorphic. Though isomorphism can be understood in many ways we will focus on continuous and completely bounded isomorphisms. Such a question of course begs the introduction of an invariant. An immediate can...

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Section: A Digression -The Purely Algebraic Casementioning

confidence: 99%

Algebras of noncommutative functions on subvarieties of the noncommutative ball: The bounded and completely bounded isomorphism problem

Salomon

Shalit

Shamovich

2020

Journal of Functional Analysis

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“…One can modify the argument to get a realization formula for B(K, L)-valued bounded nc-functions on B δ , or to prove Leech theorems (also called Toeplitzcorona theorems-see [10] and [8]). For finite-dimensional K and L, this was done in [2]; for infinite-dimensional K and L the formula was proved in [6], using results from [7].…”

Section: Closing Remarksmentioning

confidence: 99%

Global Holomorphic Functions in Several Non-Commuting Variables II

Agler¹,

McCarthy²

2018

Can. math. bull.

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“…is then an operator-valued free formal kernel in the sense of [4,3] (see also [8] which develops free Aleksandrov-Clark theory using the free formal RKHS setup). If F (Z) := α FαZ α ∈ Hnc(K), then for any α ∈ F d , the linear H−valued map defined by coefficient evaluation:…”

Section: Coefficient Evaluation and Free Formal Rkhs Let K Be An Opementioning

confidence: 99%

Column extreme multipliers of the Free Hardy Space

Jury

2019

J. London Math. Soc.

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The full Fock space over C d can be identified with the free Hardy space, H 2 (B d N ) -the unique non-commutative reproducing kernel Hilbert space corresponding to a non-commutative Szegö kernel on the non-commutative, multi-variable open unit ball B d N := ∞ n=1 C n×n ⊗ C d 1 . Elements of this space are free or non-commutative functions on B d N . Under this identification, the full Fock space is the canonical non-commutative and several-variable analogue of the classical Hardy space of the disk, and many classical function theory results have faithful extensions to this setting. In particular to each contractive (free) multiplier B of the free Hardy space, we associate a Hilbert space H(B) analogous to the deBranges-Rovnyak spaces in the unit disk, and consider the ways in which various properties of the free function B are reflected in the Hilbert space H(B) and the operators which act on it. In the classical setting, the H(b) spaces of analytic functions on the disk display strikingly different behavior depending on whether or not the function b is an extreme point in the unit ball of H ∞ (D). We show that such a dichotomy persists in the free case, where the split depends on whtether or not B is what we call column extreme. k(z, w) := 1 1 − zw * ; z, w ∈ B d , the multi-variable Szegö kernel, and zw * := z1w * 1 +...z d w * d = (w, z) C d . (Here, B d := (C d )1, the multi-variable open unit ball.) The appropriate analogue of the shift in this setting

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Noncommutative reproducing kernel Hilbert spaces

Cited by 51 publications

References 44 publications

Algebras of noncommutative functions on subvarieties of the noncommutative ball: The bounded and completely bounded isomorphism problem

Algebras of noncommutative functions on subvarieties of the noncommutative ball: The bounded and completely bounded isomorphism problem

Global Holomorphic Functions in Several Non-Commuting Variables II

Column extreme multipliers of the Free Hardy Space

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