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

Abstract

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...

Distance between reproducing kernel Hilbert spaces and geometry of finite sets in the unit ball

“…This continues a standing research program on the classification of Pick spaces and algebras; see [APV03, ARS08, DHS15, DRS11, DRS11, Har12, Har17, HL18, KMS13, MS17, OS20, SS16]. More generally, our study fits into a general theme in the geometric study of RKHSs; see, e.g., [ARSW11,CD78,McC96,Roc86] for a sample of results, or [SSS18,SSS20] for a couple of very recent examples in the noncommutative setting.…”

Non-commutative rational Clark measures

“…By the previous lemma and[61, Lemma 2.4], the closure of the joint similarity orbit of T (ζ) t contains a row contraction. Moreover, since the minimal de Branges-Rovnyak FM realization, (A, B, C, D), of b is such that A = T (0) * | M0(b) , Theorem A implies that A and hence T (0) is a pure and finite dimensional row contraction.…”

“…By [61,Lemma 2.4], the closure of the joint similarity orbit of the row d−tuple, row(Z * ), contains a d−tuple, W ′ , with norm at most W ′ row = spr(Z * ) ≤ Z row , and if Z is irreducible then its joint similarity orbit is closed so that W ′ is in the joint similarity orbit of Z * . In particular, given any ǫ > 0, Z * is jointly similar to some W ∈ C d n with W row ≤ Z row + ǫ.…”

Blaschke-Singular-Outer factorization of free non-commutative functions

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