In Memoriam

# Jacob Feldman

## Professor of Mathematics, Emeritus

UC Berkeley

1928-2017

Jacob “Jack” Feldman was born on January 10, 1928, in Philadelphia. He obtained a bachelor’s degree in mathematics from the University of Pennsylvania in 1950. He pursued graduate work at the University of Illinois but transferred after a year to the University of Chicago. There Jack came under the influence of Irving Segal and Irving Kaplansky. In 1954 he completed a dissertation under Segal entitled "Isomorphisms of Rings of Operators," which dealt with the objects now known as von Neumann algebras and their relationship to the continuous geometries of von Neumann.

Jack was then awarded a National Science Foundation postdoctoral fellowship, which he used to visit the Institute for Advanced Study for two years. For the 1956-1957 academic years, he was a visiting assistant professor at Columbia University working with Richard Kadison. Jack accepted an assistant professorship in the Department of Mathematics at the University of California, Berkeley, in 1957, was promoted to associate professor in 1960, and then to full professor in 1964. In 1969, the Department of Statistics voted to make him a 0% Professor of Statistics in recognition of his work on stochastic processes.

Jack continued to work on operator theory after his Ph.D. but also broadened his interests to probability theory, particularly Gaussian probability measures and Markov processes. One of his major results on Gaussian measures applied Kakutani’s celebrated dichotomy for infinite products of probability measures to show that two Gaussian probability measures are either mutually absolutely continuous or mutually singular and to, moreover, give a necessary and sufficient condition for mutual absolute continuity.

Jack’s most significant contributions are probably in the field of ergodic theory. He originated many fundamental ideas that shaped the development of ergodic theory and its connections with operator algebras in the last half century. His foundational papers with his colleague Calvin Moore on ergodic equivalence relations, cohomology, and von Neumann algebras are the basis of an entire theory that is still the subject of active research. His ideas were crucial in the study of actions of general locally compact groups. Jack showed how to define the basic invariant - entropy - in a novel way that has played a foundational role in all further work. He introduced a new combinatorial metric on finite strings and demonstrated its usefulness in the study of an equivalence relation on systems that arise from the various Poincaré sections of flows - Kakutani equivalence. The first results in the connections between descriptive set theory and ergodic theory, a subject that has since flourished, were due to Jack.

Jack returned to probability theory in some of his last work. One of his most notable papers from this period is a joint work with Lester Dubins, Meir Smorodinsky, and Boris Tsirelson on decreasing sequences of sigma-fields and a measure change for Brownian motion. The surprising result of this paper demonstrates that there is an error in a well-known assertion of Anatoliy Skorokhod.

Throughout his productive career, Jack supervised the doctoral work of 21 students and was an active participant in departmental affairs. He served on the Berkeley Division of the Academic Senate’s Assembly Representation (1973-74) and Graduate Council (1965-67). He retired in 1993 but continued to publish actively for several years thereafter.

Jack was an avid visual artist who left a substantial collection of paintings. Music was also essential to his life and he derived much pleasure from playing the clarinet.

Jack died on May 28, 2017. He was survived by Klara Libman, his partner of 25 years, two sons, Benjamin Feldman and David Feldman, and two granddaughters, Brooke Feldman and Ella Feldman.

Steven N. Evans

2020

Jack was then awarded a National Science Foundation postdoctoral fellowship, which he used to visit the Institute for Advanced Study for two years. For the 1956-1957 academic years, he was a visiting assistant professor at Columbia University working with Richard Kadison. Jack accepted an assistant professorship in the Department of Mathematics at the University of California, Berkeley, in 1957, was promoted to associate professor in 1960, and then to full professor in 1964. In 1969, the Department of Statistics voted to make him a 0% Professor of Statistics in recognition of his work on stochastic processes.

Jack continued to work on operator theory after his Ph.D. but also broadened his interests to probability theory, particularly Gaussian probability measures and Markov processes. One of his major results on Gaussian measures applied Kakutani’s celebrated dichotomy for infinite products of probability measures to show that two Gaussian probability measures are either mutually absolutely continuous or mutually singular and to, moreover, give a necessary and sufficient condition for mutual absolute continuity.

Jack’s most significant contributions are probably in the field of ergodic theory. He originated many fundamental ideas that shaped the development of ergodic theory and its connections with operator algebras in the last half century. His foundational papers with his colleague Calvin Moore on ergodic equivalence relations, cohomology, and von Neumann algebras are the basis of an entire theory that is still the subject of active research. His ideas were crucial in the study of actions of general locally compact groups. Jack showed how to define the basic invariant - entropy - in a novel way that has played a foundational role in all further work. He introduced a new combinatorial metric on finite strings and demonstrated its usefulness in the study of an equivalence relation on systems that arise from the various Poincaré sections of flows - Kakutani equivalence. The first results in the connections between descriptive set theory and ergodic theory, a subject that has since flourished, were due to Jack.

Jack returned to probability theory in some of his last work. One of his most notable papers from this period is a joint work with Lester Dubins, Meir Smorodinsky, and Boris Tsirelson on decreasing sequences of sigma-fields and a measure change for Brownian motion. The surprising result of this paper demonstrates that there is an error in a well-known assertion of Anatoliy Skorokhod.

Throughout his productive career, Jack supervised the doctoral work of 21 students and was an active participant in departmental affairs. He served on the Berkeley Division of the Academic Senate’s Assembly Representation (1973-74) and Graduate Council (1965-67). He retired in 1993 but continued to publish actively for several years thereafter.

Jack was an avid visual artist who left a substantial collection of paintings. Music was also essential to his life and he derived much pleasure from playing the clarinet.

Jack died on May 28, 2017. He was survived by Klara Libman, his partner of 25 years, two sons, Benjamin Feldman and David Feldman, and two granddaughters, Brooke Feldman and Ella Feldman.

Steven N. Evans

2020