His mathematical research concerned the representation theory of groups and noncommutative harmonic analysis. He began his undergraduate studies at Denison University but transferred to the University of Chicago after two years, and earned bachelor"s and master"s degrees in mathematics. After working as a military mathematical analyst, he returned to the University of Chicago, and earned his Doctor of Philosophy in 1957 with a dissertation on Fourier transformations supervised by Irving Segal.
As well as his positions at UCI and Georgia, he also worked at the Institute for Advanced Study, Massachusetts Institute of Technology, Brandeis University, and Washington University in Saint Louis.
He has over 50 academic descendants, many of them through his students Paul Sally at Brandeis and Edward North. Wilson at Washington University. With his advisor Irving Segal, Kunze was the author of the textbook Integrals and Operators (McGraw-Hill, 1968.
2nd ed, Grundlehren der Mathematischen Wissenschaften 228, Springer, 1978). With Kenneth M. Hoffman he was the author of Linear Algebra (Prentice-Hall, 1961.
2nd ed, Pearson, 1971).
In 1994, a special session on representation theory and harmonic analysis was held in honor of Kunze as part of the 889th meeting of the American Mathematical Society, and the papers from the session were published as a festschrift. In 2012, Kunze was recognized as one of the inaugural fellows of the American Mathematical Society.