Voiculescu's Theorem Research Articles

A generalization of Voiculescu's theorem for normal operators to semifinite von Neumann algebras

In this paper, we provide a generalized version of Voiculescu's theorem for normal operators by showing that, in a von Neumann algebra with separable pre-dual and a faithful, normal, semifinite, tracial weight τ, each normal operator is an arbitrarily small (max⁡{‖⋅‖,‖⋅‖2})-norm perturbation of a diagonal operator. Furthermore, in a countably decomposable, properly infinite von Neumann algebra with a faithful normal semifinite tracial weight, we prove that each self-adjoint operator can be diagonalized modulo normed ideals satisfying a natural condition. An analogue, for nuclear C⁎-algebras, of Voiculescu's absorption theorem [33, Theorem 2.4] is also proved in the case of semifinite factors.

Spectral rigidity for addition of random matrices at the regular edge

We consider the sum of two large Hermitian matrices A and B with a Haar unitary conjugation bringing them into a general relative position. We prove that the eigenvalue density on the scale slightly above the local eigenvalue spacing is asymptotically given by the free additive convolution of the laws of A and B as the dimension of the matrix increases. This implies optimal rigidity of the eigenvalues and optimal rate of convergence in Voiculescu's theorem. Our previous works [4,5] established these results in the bulk spectrum, the current paper completely settles the problem at the spectral edges provided they have the typical square-root behavior. The key element of our proof is to compensate the deterioration of the stability of the subordination equations by sharp error estimates that properly account for the local density near the edge. Our results also hold if the Haar unitary matrix is replaced by the Haar orthogonal matrix.

A relative bicommutant theorem: The stable case of Pedersen's question

In 1976, D. Voiculescu proved that every separable unital sub-C*-algebra of the Calkin algebra is equal to its (relative) bicommutant. In his minicourse (see [14]), G. Pedersen asked in 1988 if Voiculescu's theorem can be extended to a simple corona algebra of a σ-unital C*-algebra. In this note, we answer Pedersen's question for a stable σ-unital C*-algebra.

A new bicommutant theorem

We prove an analogue of Voiculescu's theorem: Relative bicommutant of a separable unital subalgebra $A$ of an ultraproduct of simple unital C*-algebras is equal to $A$.

Uniform version of Weyl–von Neumann theorem

We prove a "quantified" version of the Weyl-von Neumann theorem, more precisely, we estimate the ranks of approximants to compact operators appearing in the Voiculescu's theorem applied to commutative algebras. This allows considerable simplifications in uniform K-homology theory, namely it shows that one can represent all the uniform K-homology classes on a fixed Hilbert space with a fixed *-representation of C_0(X), for a large class of spaces X.

On the polar decomposition of circular variables

We find an elementary proof for Voiculescu's theorem on the polar decomposition of circular variables.

Voiculescu theorem, Sobolev lemma, and extensions of smooth algebras

We present the analytic foundation of a unified B-D-F extension functor Ext τ {\operatorname {Ext} _\tau } on the category of noncommutative smooth algebras, for any Fréchet operator ideal K τ {\mathcal {K}_\tau } . Combining the techniques devised by Arveson and Voiculescu, we generalize Voiculescu’s theorem to smooth algebras and Fréchet operator ideals. A key notion involved is τ \tau -smoothness, which is verified for the algebras of smooth functions, via a noncommutative Sobolev lemma. The groups Ext τ {\operatorname {Ext} _\tau } are computed for many examples.