He is the namesake of the Lumer–Phillips theorem on semigroups of operators on Banach spaces, and was the first to study L-semi-inner products.

Born in Germany and raised in France and Uruguay, he spent his professional career in the United States and Belgium.

Lumer studied at the Universidad de la República, where he came under the influence of Paul Halmos; his first mathematics paper, published in 1953, was jointly authored by Halmos and Juan Jorge Schäffer.

He completed a degree in electrical engineering at Montevideo in 1957, and traveled to Halmos' home institution, the University of Chicago, on a Guggenheim Fellowship.

He moved to the University of Mons-Hainaut in 1973, and then to the International Solvay Institutes for Physics and Chemistry in Brussels in 1999, where he remained until his death in 2005.