Bi-Boolean Independence for Pairs of Algebras

Abstract

In this paper, the notion of bi-Boolean independence for non-unital pairs of algebras is introduced thereby extending the notion of Boolean independence to pairs of algebras. The notion of B-(ℓ, r)cumulants is defined via a bi-Boolean moment-cumulant formula over the lattice of bi-interval partitions, and it is demonstrated that bi-Boolean independence is equivalent to the vanishing of mixed B-(ℓ, r)-cumulants. Furthermore, some of the simplest bi-Boolean convolutions are considered, and a bi-Boolean partial ηtransform is constructed for the study of limit theorems and infinite divisibility with respect to the additive bi-Boolean convolution. In particular, a bi-Boolean Lévy-Hinčin formula is derived in perfect analogy with the bi-free case, and some Bercovici-Pata type bijections are provided. Additional topics considered include the additive bi-Fermi convolution, some relations between the (ℓ, r)-and B-(ℓ, r)-cumulants, and bi-Boolean independence in an amalgamated setting.The last section of this paper also includes an errata that will be published with this copy of the paper. Date

Free-Boolean independence for pairs of algebras

“…We can also define bi-Boolean and bi-monotone independence with our construction. However, our definition of bi-Boolean is trivial and is different from the bi-Boolean case in [4]. On the other hand, the bi-monotone independence relation in our sense is the same as the type I bi-monotone independence in [3].…”

“…More recently, more independence relations for pairs of algebras are studied. For example in [3,4,5], conditionally bi-free independence, bi-Boolean independence, bi-monotone independence are introduced and studied.…”

Bi-monotonic Independence for Pairs of Algebras

“…If (a 1 , b 1 ) and (a 2 , b 2 ) are two such pairs which are bi-freely independent with respect to ϕ, then µ (a 1 +a 2 ,b 1 +b 2 ) depends only on µ (a 1 ,b 1 ) and µ (a 2 ,b 2 ) and is called the additive bi-free convolution of µ (a 1 ,b 1 ) and µ (a 2 ,b 2 ) , denoted µ (a 1 ,b 1 ) ⊞⊞µ (a 2 ,b 2 ) . Likewise, the notion of Boolean independence has been extended to bi-Boolean independence in [10]. However, in the case of bi-Boolean independence, ϕ need not be a state because bi-Boolean product of states is not a state in general.…”

An asymptotic property of large matrices with identically distributed Boolean independent entries

“…Boolean probability theory have been in the literature at least since early 1970's (see [22]) with various developments, from stochastic differential equations to measure theory [18]. The topic has attracted an increasing interest in the recent years, such as the works Popa and Vinnikov [15] Gu and Skoufranis [2], Jiao and Popa [3], Liu [4,5] and Popa and Hao [14]. This is the motivation for the present paper, which studies the asymptotic behavior of random matrices with independent identically distributed entries in the framework of Boolean probability.…”