A spectral radius for matrices over an operator space

With every operator space structure E on C^d, we associate a spectral radius function ρ_E on d-tuples of operators. For a d-tuple X=(X₁,…,X_d)∈M_n(C^d) of matrices we show that ρ_E(X)<1 if and only if X is jointly similar to a tuple in the open unit ball of M_n(E), that is, there is an invertible matrix S such that ‖S⁻¹XS‖_Mn(E)<1, where S⁻¹XS=(S⁻¹X₁S,…,S⁻¹X_dS). More generally, for all X∈B(K)⊗_minE we show that ρ_E(X)<1 if and only if there exists an invertible S∈B(K)⊗I such that ‖S⁻¹XS‖<1. When E is the row operator space, for example, our spectral radius coincides with the joint spectral radius considered by Bunce, Popescu, and others, and we recover the condition for a tuple of matrices to be simultaneously similar to a strict row contraction. When E is the minimal operator space min⁡(ℓ_d^∞), our spectral radius ρ_E is related to the joint spectral radius considered by Rota and Strang but differs from it and has the advantage that ρ_E(X)<1 if and only if X is simultaneously similar to a tuple of strict contractions. We show that for an nc rational function f with descriptor realization (A,b,c), the spectral radius ρ_E(A)<1 if and only if the domain of f contains a neighborhood of the noncommutative closed unit ball of the operator space dual E^⁎ of E.