An Introduction to C*-Algebras

C*-algebras were first considered in connection with quantum mechanics to model algebras of physical observables by Werner Heisenberg. Heisenberg called his new mathematics “matrix mechanics,” but the existence of the noncommutative algebra of matrices had already been noted by Jordan. Von Neumann made numerous striking discoveries later about various classes of algebras of operators on a Hilbert space. The abstract characterization of C*-algebra (Definition 1.1.7) is due to Gelfand in around 1943. The term “C*-algebra” is due to I.E. Segal in 1947, who used it to describe norm-closed *-subalgebras of bounded operators on a Hilbert space; the “C” stands for “closed.”