Can One Hear a Matrix? Recovering a Real Symmetric Matrix from Its Spectral Data

The spectrum of a real and symmetric $$N\times N$$ N × N matrix determines the matrix up to unitary equivalence. More spectral data is needed together with some sign indicators to remove the unitary ambiguities. In the first part of this work, we specify the spectral and sign information required for a unique reconstruction of general matrices. More specifically, the spectral information consists of the spectra of the N nested main minors of the original matrix of the sizes $$1,2,\ldots ,N$$ 1 , 2 , … , N . However, due to the complicated nature of the required sign data, improvements are needed in order to make the reconstruction procedure feasible. With this in mind, the second part is restricted to banded matrices where the amount of spectral data exceeds the number of the unknown matrix entries. It is shown that one can take advantage of this redundancy to guarantee unique reconstruction of generic matrices; in other words, this subset of matrices is open, dense and of full measure in the set of real, symmetric and banded matrices. It is shown that one can optimize the ratio between redundancy and genericity by using the freedom of choice of the spectral information input. We demonstrate our constructions in detail for pentadiagonal matrices.

A decomposition of the tensor product of matrices

In this paper we decompose (under unitary equivalence) the tensor product A ? A into a direct sum of irreducible matrices, when A is a 3 x 3 matrix.

On Ultraproduct Embeddings and Amenability for Tracial von Neumann Algebras

We define the notion of self-tracial stability for tracial von Neumann algebras and show that a tracial von Neumann algebra satisfying the Connes embedding problem (CEP) is self-tracially stable if and only if it is amenable. We then generalize a result of Jung by showing that a separable tracial von Neumann algebra that satisfies the CEP is amenable if and only if any two embeddings into $R^{\mathcal{U}}$ are ucp-conjugate. Moreover, we show that for a II$_1$ factor $N$ satisfying CEP, the space $\mathbb{H}$om$(N, \prod _{k\to \mathcal{U}}M_k)$ of unitary equivalence classes of embeddings is separable if and only $N$ is hyperfinite. This resolves a question of Popa for Connes embeddable factors. These results hold when we further ask that the pairs of embeddings commute, admitting a nontrivial action of $\textrm{Out}(N\otimes N)$ on ${\mathbb{H}}\textrm{om}(N\otimes N, \prod _{k\to \mathcal{U}}M_k)$ whenever $N$ is non-amenable. We also obtain an analogous result for commuting sofic representations of countable sofic groups.

Strongly Peaking Representations and Compressions of Operator Systems

We use Arveson’s notion of strongly peaking representation to generalize uniqueness theorems for free spectrahedra and matrix convex sets that admit minimal presentations. A fully compressed separable operator system necessarily generates the $C^*$-envelope and is such that the identity is the direct sum of strongly peaking representations. In particular, a fully compressed presentation of a separable operator system is unique up to unitary equivalence. Under various additional assumptions, minimality conditions are sufficient to determine a separable operator system uniquely.