English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

The key result in the paper concerns two transformations, Phi(rho, psi) andB_t(psi) on states on the algebra of non-commutative polynomials, orequivalently on joint distributions of d-tuples of non-commuting operators.These transformations are related to free probability: Phi intertwines theaction of B_t and the free convolution with the semigroup {rho_t}. The maps{B_t} were introduced by Belinschi and Nica as a semigroup of transformationssuch that B_1 is the bijection between infinitely divisible distributions inthe Boolean and free probability theories. They proved the intertwiningproperty above for a single-variable version of the map Phi and the particularcase of the free heat semigroup. The more general two-variable map Phi comes,not from free probability, but from the theory of two-state algebras, alsocalled the conditionally free probability theory, introduced by Bozejko,Leinert, and Speicher. Orthogonality of the c-free versions of the Appellpolynomials, investigated in arXiv:0803.4279, is closely related to thesingle-variable map Phi. On the other hand, more general free Meixner familiesbehave well under all the transformations above, and provide clues to theirgeneral behavior. Besides the evolution equation, other results include thepositivity of the map Phi and descriptions of its fixed points and range.

Free evolution on algebras with two states