Let {ak}k=1 be free identically distributed positive non–commuting random variables with probability measure distribution μ. In this paper we proved a multiplicative version of the Free Central Limit Theorem. More precisely, let bn = a 1/2 1 a 1/2 2 . . . an . . . a 1/2 2 a 1/2 1 then bn is a positive operator with the same moments as xn = a1a2 . . . an and b 1/2n n converges in distribution to...

We explain how multiplicative bundle gerbes over a compact, connected and simple Lie group G lead to a certain fusion category of equivariant bundle gerbe modules given by pre-quantizable Hamiltonian LG-manifolds arising from Alekseev-Malkin-Meinrenken’s quasiHamiltonian G-spaces. The motivation comes from string theory namely, by generalising the notion of D-branes in G to allow subsets of G t...

Let {Tk}k=1 be a family of ∗–free identically distributed operators in a finite von Neumann algebra. In this work we prove a multiplicative version of the free central limit Theorem. More precisely, let Bn = T ∗ 1 T ∗ 2 . . . T ∗ nTn . . . T2T1 then Bn is a positive operator and B 1/2n n converges in distribution to an operator Λ. We completely determine the probability distribution ν of Λ from...

For a Poincaré duality space X and a map X → B, consider the homotopy fiber product X × X . If X is orientable with respect to a multiplicative cohomology theory E, then, after suitably regrading, it is shown that the E-homology of X × X has the structure of a graded associative algebra. When X → B is the diagonal map of a manifold X , one recovers a result of Chas and Sullivan about the homolo...

Using concepts of noncommutative probability we show that the Löwner’s evolution equation can be viewed as providing a map from paths of measures to paths of probability measures. We show that the fixed point of the Löwner map is the convolution semigroup of the semicircle law in the chordal case, and its multiplicative analogue in the radial case. We further show that the Löwner evolution “spr...

‎We extend the notion of approximately multiplicative to approximately n-multiplicative maps between locally multiplicatively convex algebras and study some properties of these maps‎. ‎‎W‎e prove that every approximately n-multiplicative linear functional on a functionally continuous locally multiplicatively convex algebra is continuous‎. ‎We also study the relationship between approximately mu...

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Semi-invertible multiplicative ergodic theorems establish the existence of an Oseledets splitting for cocycles of non-invertible linear operators (such as transfer operators) over an invertible base. Using a constructive approach, we establish a semi-invertible multiplicative ergodic theorem that for the first time can be applied to the study of transfer operators associated to the composition ...

Our goal will be to find subsets of Zm that form groups under the operation of multiplication modulo m. By utilizing the isomorphism Zm ∼= Zn ⊕ Zk, we will find multiplicative groups contained in Zn ⊕ Zk and then map these back to Zm. In particular, if m = nk with gcd(n, k) = 1, our objective is to find particular multiplicative subsets of Zn × Zk that are groups and whose first coordinate is a...

There are several kinds of hyperrings, for example, Krasnerhyperrings, multiplicative hyperring, general hyperrings and$H_v$-rings. In a multiplicative hyperring, the multiplication isa hyperoperation, while the addition is a binary operation. In this paper, the notion of derivation on multiplicative hyperrings is introduced and some related properties are investigated. {bf Keywords:} multiplic...