Operator algebra

About: Operator algebra is a research topic. Over the lifetime, 5783 publications have been published within this topic receiving 165303 citations.

Grassmannian coset models and unitary representations of w

Abstract: It is shown that the 2 – d coset models SU(p + 1)N/SU(p)N ⊗ U(1) provide unitary representations of the chiral operator algebra W∞ in the large level (N → ∞) limit, with central charge c = 2p. For p ≥ 2, the corresponding field theories possess additional symmetries which given rise to a U(p) matrix generalization of W∞, denoted by $W_\infty^p$. Its commutation relations are obtained in closed form for all values of p and W∞ is identified with the U(1) trace part of $W_\infty^p$. It is also shown that $W_\infty^p$ at large p is associated with the algebra of symplectic diffeomorphisms in four dimensions.

Operator-valued distributions: I. Characterizations of freeness

Abstract: Let $M$ be a $B$-probability space. Assume that $B$ itself is a $D$-probability space; then $M$ can be viewed as $D$-probability space as well. Let $X$ be in $M$. We look at the question of relating the properties of $X$ as $B$-valued random variable to its properties as $D$-valued random variable. We characterize freeness of $X$ from $B$ with amalgamation over $D$: (a) in terms of a certain factorization condition linking the $B$-valued and $D$-valued cumulants of $X$, and (b) for $D$ finite-dimensional, in terms of linking the $B$-valued and the $D$-valued Fisher information of $X$. We give an application to random matrices. For the second characterization we derive a new operator-valued description of the conjugate variable and introduce an operator-valued version of the liberation gradient.