In the framework of deformation quantization we apply the formal GNS construction to find representations of the deformed algebras in pre-Hilbert spaces over C[[λ]] and establish the notion of local operators in these pre-Hilbert spaces. The commutant within the local operators is used to distinguish ‘thermal’ from ‘pure’ representations. The computation of the local commutant is exemplified in...

A left Bol loop is a loop satisfying x(y(xz)) = (x(yx))z. The commutant of a loop is the set of all elements which commute with all elements of the loop. In a finite Bol loop of odd order or of order 2k, k odd, the commutant is a subloop. We investigate conditions under which the commutant of a Bol loop is not a subloop. In a finite Bol loop of order relatively prime to 3, the commutant generat...

In thermal states of chiral theories, as recently investigated by H.-J. Borchers and J. Yngvason, there exists a rich group of hidden symmetries. Here we show that this leads to a radical converse of of the Hawking-Unruh observation in the following sense. The algebraic commutant of the algebra associated with a (heat bath) thermal chiral system can be used to reprocess the thermal system into ...

Let T (QC) (resp. T ) be the C∗-algebra generated by the Toeplitz operators {Tφ : φ ∈ QC} (resp. {Tφ : φ ∈ L∞}) on the Hardy space H2 of the unit circle. A well-known theorem of Davidson asserts that T (QC) is the essential commutant of T . We show that the essential commutant of T (QC) is strictly larger than T . Thus the image of T in the Calkin algebra does not satisfy the double commutant r...

In this paper, we focus on the structured multi-frame vectors in Hilbert $C^*$-modules. More precisely, it will be shown that the set of all complete multi-frame vectors for a unitary system can be parameterized by the set of all surjective operators, in the local commutant. Similar results hold for the set of all complete wandering vectors and complete multi-Riesz vectors, when the surjective ...

ABSTRACT. In this paper, using the group-like property of local inverses of a finite Blaschke product φ, we will show that the largest C-algebra in the commutant of the multiplication operator Mφ by φ on the Bergman space is finite dimensional, and its dimension equals the number of connected components of the Riemann surface of φ ◦ φ over the unit disk. If the order of the Blaschke product φ i...

In order to describe non-Hamiltonian (dissipative) systems in quantum theory we need to use non-Lie algebra such that commutators of this algebra generate Lie subalgebra. It was shown that classical connection between analytic group (Lie group) and Lie algebra, proved by Lie theorems, exists between analytic loop, commutant of which is associative subloop (group), and commutant Lie algebra (an ...

Collective rotation channels are a fundamental class of channels in quantum computing and quantum information theory. The commutant of the noise operators for such a channel is a C∗-algebra which is equal to the set of fixed points for the channel. Finding the precise spatial structure of the commutant algebra for a set of noise operators associated with a channel is a core problem in quantum e...

Analogue to commutants in the theory of associative algebras, one can construct a new subalgebra of vertex algebra known as a vertex algebra commutant. In this paper, for the adjoint representation V of Lie algebra sl(2,C), we describe a commutant of βγSystem S(V ) by giving its generators, moreover, we get a new Howe pair of vertex algebras.