We study the free probabilistic analog of optimal couplings for quadratic cost, where classical probability spaces are replaced by tracial von Neumann algebras, and measures on $\mathbb{R}^m$ non-commutative laws $m$-tuples. prove an Monge-Kantorovich duality which characterizes with respect to Biane Voiculescu's $L^2$-Wasserstein distance using a new type convex functions. As consequence, we s...

Suppose F is a finite set of selfadjoint elements in a tracial von Neumann algebra M . For α > 0, F is α-bounded if Pα(F ) < ∞ where Pα is the α-packing entropy of F introduced in [7]. We say that M is strongly 1-bounded if M has a 1-bounded finite set of selfadjoint generators F such that there exists an x ∈ F with χ(x) > −∞. It is shown that if M is strongly 1-bounded, then any finite set of ...

We introduce a notion of amenable normal extension S unital ring R with finite approximation system F, encompassing the algebras over field Gromov and Elek, twisted crossed product by an group, tensor extension. It is shown that every Sylvester matrix rank function rk preserved has canonical to rk_F for S. In case extension, it also depends on continuously. When group action C*-algebra preservi...

in a commutative noetherian local complex normed algebra which is complete in its m-adic metric there are only finitely many closed prime ideals.

Specht's Theorem states that two matrices A and B in M n ( C ) are unitarily equivalent if only tr w , ? = for all words x y non-commuting variables . In this article we examine to what extent trace condition characterises approximate unitary equivalence uniformly hyperfinite (UHF) -algebras. particular, show given elements a b of the universal UHF-algebra Q which generate -algebras satisfying ...

Assume that A is a unital separable simple C*-algebra with real rank zero, stable rank one, strict comparison of projections, and that its tracial simplex T(A) has a finite number of extremal points. We prove that every self-adjoint element a in A with τ(a) = 0 for all τ ∈ T(A) is the sum of two commutators in A and that that every positive element of A is a linear combination of projections wi...

Let A and B be unital separable simple amenable C*-algebras which satisfy the Universal Coefficient Theorem. Suppose that A and B are Z-stable and are of rationally tracial rank no more than one. We prove the following: Suppose that φ, ψ : A → B are unital *monomorphisms. There exists a sequence of unitaries {un} ⊂ B such that lim n→∞ unφ(a)un = ψ(a) for all a ∈ A, if and only if [φ] = [ψ] in K...

In a commutative Noetherian local complex normed algebra which is complete in its M-adic metric there are only finitely many closed prime ideals.