TY - JOUR

T1 - Algebras of noncommutative functions on subvarieties of the noncommutative ball

T2 - The bounded and completely bounded isomorphism problem

AU - Salomon, Guy

AU - Shalit, Orr M.

AU - Shamovich, Eli

N1 - Funding Information:
The first author was partially supported by the Clore Foundation. The second author was partially supported by Israel Science Foundation Grant No. 195/16. The third author was partially supported by the Fields Institute for Research in the Mathematical Sciences.
Publisher Copyright:
© 2019 Elsevier Inc.

PY - 2020/4/15

Y1 - 2020/4/15

N2 - Given a noncommutative (nc) variety V in the nc unit ball Bd, 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 pseudo-hyperbolic 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∞(Bd) studied by Davidson–Pitts and by Popescu. In particular, we find that Aut(H∞(Bd)) 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.

AB - Given a noncommutative (nc) variety V in the nc unit ball Bd, 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 pseudo-hyperbolic 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∞(Bd) studied by Davidson–Pitts and by Popescu. In particular, we find that Aut(H∞(Bd)) 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.

KW - Noncommutative analysis

KW - Noncommutative analytic geometry

KW - Noncommutative functions

KW - Operator algebras

UR - http://www.scopus.com/inward/record.url?scp=85077375519&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2019.108427

DO - 10.1016/j.jfa.2019.108427

M3 - Article

AN - SCOPUS:85077375519

SN - 0022-1236

VL - 278

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 7

M1 - 108427

ER -