Probabilistic Operator Algebra Seminar: Boolean probability basics
Seminar | January 22 | 2-4 p.m. | 736 Evans Hall
Jorge Garza Vargas, UC Berkeley
With the introduction of free independence by D.V. Voiculescu, it became clear that in the framework of non-commutative probability there are other notions of independence different than that of (classical) independence. In 1997, R. Speicher defined a notion of universal product for which he showed that there are three types of independence. In the category of unital algebras the tensor and free independence are the only existing ones. On the other hand when algebras are not required to have a unit, the product provided by Boolean independence is also admitted as a universal product. The Boolean convolution between measures was formally introduced by Speicher and R. Woroudi in 1993, although it had previously appeared in the literature in different contexts, for example, as partial cumulants in stochastic differential equations. Later, in 2006, H. Bercovici provided the product for Hilbert spaces that, in the context of operator algebras, corresponds to the Boolean convolution between measures. In this talk we will survey the basics of Boolean probability together with some results that show the similarities and differences that it has with the classical theory of probability.