We introduce a new class of operator algebras on Hilbert space. To each bounded linear operator a spectral algebra is associated. These algebras are quite substantial, each containing the commutant of the associated operator, frequently as a proper subalgebra. We establish several sufficient conditions for a spectral algebra to have a nontrivial invariant subspace. When the associated operator ...

We develop a structure theory for decoherence-free subspaces and noiseless subsystems that applies to arbitrary (not necessarily unital) quantum operations. The theory can be alternatively phrased in terms of the superoperator perspective, or the algebraic noise commutant formalism. As an application, we propose a method for finding all such subspaces and subsystems for arbitrary quantum operat...

We consider two twisted actions of a countable discrete group on $\sigma$-unital $C^*$-algebras. Then by taking the reduced crossed products, we get inclusions suppose that they are strongly Morita equivalent as Also, one $C^*$-algebras is irreducible, is, relative commutant $C^*$-algebra in multiplier product trivial. show then up to some automorphism group.

We apply Hilbert module methods to show that normal completely positive maps admit weak tensor dilations. Appealing to a duality between weak tensor dilations and extensions of CP-maps, we get an existence proof for certain extensions. We point out that this duality is part of a far reaching duality between a von Neumann bimodule and its commutant in which also other dualities, known and new, f...

A complex number λ is an extended eigenvalue of an operator A if there is a nonzero operator X such that AX = λXA. We characterize the the set of extended eigenvalues for operators acting on finite dimensional spaces, finite rank operators, Jordan blocks, and C0 contractions. We also describe the relationship between the extended eigenvalues of an operator A and its powers. We derive some appli...

We construct a series of inclusions of hyperfinite subfactors Pi in the centralizer R of Powers’ IIIt factor, Pi+1 ⊂ Pi ⊂ R. It is proven that: ∀i ∈ N, P ′ i ∩R = C, [Pi : Pi+1] ≥ λ = (1 + t) t Via an outer action of Z2 on R, we also construct an inclusion of hyperfinite II1 factors, M3 ⊂ M1 with the index equal to λ, such that the relative commutant M1 ∩M ′ 3 is four dimensional.

Motivated by the multivariate wavelet theory, and by the spectral theory of transfer operators, we construct an abstract affine structure and a multiresolution associated to a matrix-valued weight. We describe the one-to-one correspondence between the commutant of this structure and the fixed points of the transfer operator. We show how the covariant representation can be realized on Rn if the ...

A bi-unitary connection in subfactor theory of Jones producing a finite depth gives 4-tensor appearing recent work Bultinck-Mariena-Williamson-Sahinoglu-Haegemana-Verstraete on 2-dimensional topological order and anyons. In their work, they have special projection called projector matrix product operator. We prove that the range this length k is naturally identified with k-th higher relative co...