- Non-commutative Lebesgue decomposition of non-commutative measures (arXiv)
Author: Fuad Naderi
Summary: A positive noncommutative (NC) measure is a positive linear function on a free disk operator system produced by a d-tuple of noncommutative isometry. We construct a natural Lebesgue decomposition for positive NC measures relative to other such measures by introducing a hybrid form, its Cauchy transform, and the technique of NC reproduction kernel Hilbert space (RKHS). Our study is an extension of the Jury-Martin decomposition that originally decomposed the positive NC measure against the standard NC Lebesgue measure. In fact, it provides a more generalized definition of absolute continuity and singularity. This comes down to the definition when the split measure is a standard NC Lebesgue measure. This generalized definition makes it possible to extend the Jury-Martin theory for arbitrary split NC measures, recovering the decomposition when the split NC measure is a Lebesgue measure. Our study implies a Lebesgue decomposition for the representation of the Kunz-Toeplitz C* algebra. Moreover, our RKHS method gives a new proof of the classical Lebesgue decomposition when applied to the classical one-dimensional setting, i.e. d = 1.
2. Representation of mapping of semi-bounded form and its Lebesgue-type decomposition (arXiv)
Author: Seppo Hassi, Henk de Snou
Summary: For a semibounded sesquilinear form t in a Hilbert space H, there exists a representation mapping Q from H to another Hilbert space K.[φ,ψ]−c(φ,ψ)=(Qφ,Qψ), φ,ψ∈domt, c∈R is the lower bound of t. The map representation provides a simplification tool for studying general semi-bounded forms. By representing the map, the closedness, closure, and singularity of t are immediately translated into the corresponding properties of the operator Q, and vice versa. Also, the properties of the sum decomposition t=t1+t2 between a non-negative form t and two other non-negative forms t1 and t2 in H can be analyzed by the associated non-negative reduction K∈B(K). This helps, for example, to establish an explicit operator theory characterization of whether the addends t1 and t2 are mutually singular. Such a sum decomposition is used to study the properties of the so-called Lebesgue-type decomposition of the semibounded form t. Here t1 is closable and t2 is singular. In particular, this includes the Lebesgue decomposition of semi-bounded forms by B. Simon. Furthermore, for a semi-bounded form t with a representation mapping Q, the corresponding semi-bounded self-adjoint relation Q∗Q∗∗+c is uniquely defined by the limit version of the classical representation theorem for the form t studied by W. is shown to be determined. Arendt and T. ter Erst by field. Representing the map completely handles the convergence of monotonic sequences in semibounded form.
