• 제목/요약/키워드: probability algebras

검색결과 7건 처리시간 0.023초

ℂ-VALUED FREE PROBABILITY ON A GRAPH VON NEUMANN ALGEBRA

  • Cho, Il-Woo
    • 대한수학회지
    • /
    • 제47권3호
    • /
    • pp.601-631
    • /
    • 2010
  • In [6] and [7], we introduced graph von Neumann algebras which are the (groupoid) crossed product algebras of von Neumann algebras and graph groupoids via groupoid actions. We showed that such crossed product algebras have the graph-depending amalgamated reduced free probabilistic properties. In this paper, we will consider a scalar-valued $W^*$-probability on a given graph von Neumann algebra. We show that a diagonal graph $W^*$-probability space (as a scalar-valued $W^*$-probability space) and a graph W¤-probability space (as an amalgamated $W^*$-probability space) are compatible. By this compatibility, we can find the relation between amalgamated free distributions and scalar-valued free distributions on a graph von Neumann algebra. Under this compatibility, we observe the scalar-valued freeness on a graph von Neumann algebra.

Multinomial Probability Distribution and Quantum Deformed Algebras

  • Fridolin Melong
    • Kyungpook Mathematical Journal
    • /
    • 제63권3호
    • /
    • pp.463-484
    • /
    • 2023
  • An examination is conducted on the multinomial coefficients derived from generalized quantum deformed algebras, and on their recurrence relations. The 𝓡(p, q)-deformed multinomial probability distribution and the negative 𝓡(p, q)-deformed multinomial probability distribution are constructed, and the recurrence relations are determined. From our general result, we deduce particular cases that correspond to quantum algebras considered in the literature.

CONDITIONAL EXPECTATIONS GENERATING THE COMMUTANTS OF SUBALGEBRAS OF $L^{\infty}$

  • Lambert, Alan
    • 대한수학회지
    • /
    • 제36권4호
    • /
    • pp.699-705
    • /
    • 1999
  • Given a probability space and a subsigma algebra A, each measure equivalent to the probability measure generates a different conditional expectation operator. We characterize those which act boundedly on the original $L^2$ space, and show there are sufficiently many such conditional expectations to generate the commutant of $L^{\infty}$ (A).

  • PDF

SATURATED STRUCTURES FROM PROBABILITY THEORY

  • Song, Shichang
    • 대한수학회지
    • /
    • 제53권2호
    • /
    • pp.315-329
    • /
    • 2016
  • In the setting of continuous logic, we study atomless probability spaces and atomless random variable structures. We characterize ${\kappa}$-saturated atomless probability spaces and ${\kappa}$-saturated atomless random variable structures for every infinite cardinal ${\kappa}$. Moreover, ${\kappa}$-saturated and strongly ${\kappa}$-homogeneous atomless probability spaces and ${\kappa}$-saturated and strongly ${\kappa}$-homogeneous atomless random variable structures are characterized for every infinite cardinal ${\kappa}$. For atomless probability spaces, we prove that ${\aleph}_1$-saturation is equivalent to Hoover-Keisler saturation. For atomless random variable structures whose underlying probability spaces are Hoover-Keisler saturated, we prove several equivalent conditions.

FREE PROBABILITY THEORY AND ITS APPLICATION

  • Heo, Jaeseong
    • 충청수학회지
    • /
    • 제15권2호
    • /
    • pp.13-23
    • /
    • 2003
  • We prove a simplicity of the $C^*$-algebra generated by some $C^*$-subalgebra and a Haar unitary in a free product of finite von Neumann algebras. Some examples and questions are given.

  • PDF

RELATIONS BETWEEN THE ITO PROCESSES

  • Choi, Won
    • 대한수학회논문집
    • /
    • 제10권1호
    • /
    • pp.207-213
    • /
    • 1995
  • Let $(\Omega, F, P)$ be a probability space with F a $\sigma$-algebra of subsets of the measure space $\Omega$ and P a probability measure on $\Omega$. Suppose $a > 0$ and let $(F_t)_{t \in [0,a]}$ be an increasing family of sub-$\sigma$-algebras of F. If $r > 0$, let $J = [-r,0]$ and $C(J, R^n)$ the Banach space of all continuous paths $\gamma : J \to R^n$ with the sup-norm $\Vert \gamma \Vert = sup_{s \in J}$\mid$\gamma(s)$\mid$$ where $$\mid$\cdot$\mid$$ denotes the Euclidean norm on $R^n$. Let E,F be separable real Banach spaces and L(E,F) be the Banach space of all continuous linear maps $T : E \to F$.

  • PDF

A NOTE ON ITO PROCESSES

  • Park, Won
    • 대한수학회논문집
    • /
    • 제9권3호
    • /
    • pp.731-737
    • /
    • 1994
  • Let $(\Omega, F, P)$ be a probability space with F a $\sigma$-algebra of subsets of the measure space $\Omega$ and P a probability measures on $\Omega$. Suppose $a > 0$ and let $(F_t)_{t \in [0,a]}$ be an increasing family of sub-$\sigma$- algebras of F. If $r > 0$, let $J = [-r, 0]$ and $C(J, R^n)$ the Banach space of all continuous paths $\gamma : J \to R^n$ with the sup-norm $\Vert \gamma \Vert_C = sup_{s \in J} $\mid$\gamma(x)$\mid$$ where $$\mid$\cdot$\mid$$ denotes the Euclidean norm on $R^n$. Let E and F be separable real Banach spaces and L(E,F) be the Banach space of all continuous linear maps $T : E \to F$ with the norm $\Vert T \Vert = sup {$\mid$T(x)$\mid$_F : x \in E, $\mid$x$\mid$_E \leq 1}$.

  • PDF