The Isomorphism Problem for Complete Pick Algebras: A Survey

Salomon, Guy; Shalit, Orr

doi:10.1007/978-3-319-31383-2_9

Cited by 10 publications

(25 citation statements)

References 27 publications

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“…Proof. We import the "disc trick" from [23] to the current setting (see also [74,Lemma 5.9]). Let G be a conformal equivalence mapping V onto W. If 0 is mapped by G to 0, we are done.…”

Section: The Isomorphism Problem For Homogeneous Varietiesmentioning

confidence: 99%

“…In fact Davidson, Ramsey and Shalit in [23] and Hartz in [35] proved that if d < ∞, then for homogeneous varieties V and W we have that M V ∼ = M W algebraically if and only if there exists a linear map ϕ ∈ GL d (C), such that ϕ(V ) = W . We refer the reader to [74] for a detailed survey and also additional results on these questions.…”

Section: Introductionmentioning

confidence: 99%

“…We call a subset V ⊆ B d a variety if V is the joint zero set of a family of functions in M d . In [19,23,24,35,36,45,70] (see also the survey paper [74]), the following problem was investigated. For a variety V ⊆ B d , consider the Hilbert space F V = span{k v : v ∈ V }, and define M V = Mult F V .…”

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confidence: 99%

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Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball

Salomon

Shalit

Shamovich

2018

Trans. Amer. Math. Soc.

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We study algebras of bounded, noncommutative (nc) analytic functions on nc subvarieties of the nc unit ball. Given an nc variety V in the nc unit ball B d , we identify the algebra of bounded analytic functions on V -denoted H ∞ (V) -as the multiplier algebra Mult H V of a certain reproducing kernel Hilbert space H V consisting of nc functions on V. We find that every such algebra H ∞ (V) is completely isometrically isomorphic to the quotient H ∞ (B d )/J V of the algebra of bounded nc holomorphic functions on the ball by the ideal J V of bounded nc holomorphic functions which vanish on V. In order to demonstrate this isomorphism, we prove that the space H V is an nc complete Pick space (a fact recently proved -by other methods -by Ball, Marx and Vinnikov).We investigate the problem of when two algebras H ∞ (V) and H ∞ (W) are (completely) isometrically isomorphic. If the variety W is the image of V under an nc analytic automorphism of B d , then H ∞ (V) and H ∞ (W) are completely isometrically isomorphic. We prove that the converse holds in the case where the varieties are homogeneous; in general we can only show that if the algebras are completely isometrically isomorphic, then there must be nc holomorphic maps between the varieties (in the case d = ∞ we need to assume that the isomorphism is also weak- * continuous).We also consider similar problems regarding the bounded analytic functions that extend continuously to the boundary of B d and related norm closed algebras; the results in the norm closed setting are somewhat simpler and work for the case d = ∞ without further assumptions.Along the way, we are led to consider some interesting problems on function theory in the nc unit ball. For example, we study various versions of the Nullstellensatz (that is, the problem of to what extent an ideal is determined by its zero set), and we obtain perfect Nullstellensatz in both the homogeneous as well as the commutative cases.Analogues of the Nevanlinna-Pick interpolation on the noncommutative ball first appeared in [21] and [12]. More general noncommutative versions of the classical interpolation and realization results appeared recently in the works of Agler and M c Carthy [5] and Ball, Marx and Vinnikov [14,15], who also introduced a generalization of reproducing kernel Hilbert spaces to the free setting.Our first goal in this work is to show that the full Fock space is a noncommutative reproducing kernel Hilbert space (nc RKHS), and its algebra of multipliers is, on the one hand, the algebra of bounded functions on the noncommutative ball (such that the multiplier norm and the supremum norm coincide), and, on the other hand, that this algebra coincides with the WOT closed algebra considered by Arias-Popescu and Davidson-Pitts. With this identification in hand, our second goal is to show that several results of [14] in the case of the noncommutative ball follow from established operator algebraic techniques and results, in particular, the complete Pick property of the noncommutative kernel of the full Fock space.W...

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“…Proof. We import the "disc trick" from [23] to the current setting (see also [74,Lemma 5.9]). Let G be a conformal equivalence mapping V onto W. If 0 is mapped by G to 0, we are done.…”

Section: The Isomorphism Problem For Homogeneous Varietiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball

Salomon

Shalit

Shamovich

2018

Trans. Amer. Math. Soc.

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“…Proof. We can import the "disc trick" used in [13,Proposition 4.7] to the current setting (see also [42,Lemma 5.9]). Since the argument is just a couple of paragraphs long, we include it for completeness.…”

Section: The Homogeneous Casementioning

confidence: 99%

“…We are now ready to study the isomorphism problem for the algebras of bounded analytic functions on commutative nc varieties. This problem was treated extensively in the fully commutative case, that is, when the algebra H ∞ (V) lives on a variety V which is minimal, in the sense that it is the minimal nc variety in B d which contains V = V(1); this is referred to as the isomorphism problem for complete Pick algebras, see [10,13,14,21,27,28,42]. In [43,Section 11] we explained how this problem can be investigated in the nc commutative setting.…”

Section: The Commutative 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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An Invitation to the Drury–Arveson Space

Hartz

2023

Fields Institute Monographs

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The Isomorphism Problem for Complete Pick Algebras: A Survey

Cited by 10 publications

References 27 publications

Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball

Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball

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

An Invitation to the Drury–Arveson Space

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