Faithfulness of bifree product states

Ramsey, Christopher

doi:10.1090/proc/14194

Cited by 4 publications

(4 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This is a setting which includes many canonical examples and thus is of great interest. Note we will not assume that ϕ is tracial on A nor faithful on A as these properties need not occur in most bi-free systems (see [2] and [23] respectively).…”

Section: Definition and Basic Propertiesmentioning

confidence: 99%

See 1 more Smart Citation

Analogues of Entropy in Bi-Free Probability Theory: Microstates

Charlesworth

Skoufranis

2021

International Mathematics Research Notices

View full text Add to dashboard Cite

In this paper, we extend the notion of microstate free entropy to the bi-free setting. In particular, using the bi-free analogue of random matrices, microstate bi-free entropy is defined. Properties essential to an entropy theory are developed, such as the behaviour of the entropy when transformations on the left variables or on the right variables are performed. In addition, the microstate bi-free entropy is demonstrated to be additive over bi-free collections provided additional regularity assumptions are included and is computed for all bi-free central limit distributions. Moreover, an orbital version of bi-free entropy is examined, which provides a tighter upper bound for the subadditivity of microstate bi-free entropy and provides an alternate characterization of bi-freeness in certain settings.

show abstract

Section: Definition and Basic Propertiesmentioning

confidence: 99%

“…the free case). Of course, this is one reason why the states in a left-right, tracially bi-partite, C * -non-commutative probability space need not be faithful as the work of [23] shows we would be greatly restricting the systems we can study in that the bi-free product of faithful states need not be faithful.…”

Section: A Characterization Of Bi-freenessmentioning

confidence: 99%

Analogues of Entropy in Bi-Free Probability Theory: Microstates

Charlesworth

Skoufranis

2021

International Mathematics Research Notices

View full text Add to dashboard Cite

show abstract

“…We will call (A, ϕ) a C * -non-commutative probability space. Note we will assume neither that ϕ is tracial nor faithful on A as these properties need not occur in most bi-free systems (see [1,Theorem 6.1] and [16] respectively). Let L 2 (A, ϕ) denote the GNS Hilbert space induced from the sesquilinear form…”

Section: Bi-free Difference Quotients and Conjugate Variablesmentioning

confidence: 99%

Analogues of Entropy in Bi-Free Probability Theory: Non-Microstate

Charlesworth¹,

Skoufranis²

2019

Preprint

View full text Add to dashboard Cite

In this paper, we extend the notion of non-microstate free entropy to the bi-free setting. Using a diagrammatic approach involving bi-non-crossing diagrams, bi-free difference quotients are constructed as analogues of the free partial derivations. Adjoints of bi-free difference quotients are discussed and used to define bi-free conjugate variables. Notions of bi-free Fisher information, non-microstate bi-free entropy, and non-microstate bi-free entropy dimension are defined and known properties in the free setting are extended to the bi-free setting.

show abstract

“…This is a setting that many canonical examples fit into and thus is of great interest. Note we will not assume that ϕ is tracial on A nor faithful on A as these properties need not occur in most bi-free systems (see [2] and [13] respectively). Using the fact that the system is bi-partite, the definition of Γ R (X 1 , .…”

Section: Definition and Basic Propertiesmentioning

confidence: 99%