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

Abstract

Given a noncommutative (nc) variety $\mathfrak{V}$ in the nc unit ball ${\mathfrak{B}}_d$, we consider the algebra $H^{\infty }(\mathfrak{V})$ of bounded nc holomorphic functions on $\mathfrak{V}$. We investigate the problem of when two algebras $H^{\infty }(\mathfrak{V})$ and $H^{\infty }(\mathfrak{W})$ are isomorphic. We prove that these algebras are weak-$\ast$ continuously isomorphic if and only if there is an nc biholomorphism $G:\tilde{\mathfrak{W}}\to \tilde{\mathfrak{V}}$ between the similarity envelopes that is bi-Lipschitz with respect to the free pseudo-hyperbolic metric. Moreover, such an isomorphism always has the form $f\mapsto f\circ G$, where $G$ is an nc biholomorphism. These results also shed some new light on automorphisms of the noncommutative analytic Toeplitz algebras $H^{\infty }({\mathfrak{B}}_d)$ studied by Davidson--Pitts and by Popescu. In particular, we find that $\mathrm{Aut}(H^{\infty }({\mathfrak{B}}_d))$ is a proper subgroup of $\mathrm{Aut}({\tilde{\mathfrak{B}}}_d)$. When $d<\infty$ and the varieties are homogeneous, we remove the weak-$\ast$ 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.

45 pages. Some details were added and more minor changes

References in corpus (4)

• Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball
• The isomorphism problem for some universal operator algebras
• On fixed points of self maps of the free ball
• Noncommutative hyperbolic metrics

Cited by in corpus (4)

• von Neumann's inequality for row contractive matrix tuples
• Rational Cuntz states peak on the free disk algebra
• Tensor algebras of subproduct systems and noncommutative function theory
• Distance between reproducing kernel Hilbert spaces and geometry of finite sets in the unit ball