A categorial-algebraic description of $K$-theory for $C^*$-algeras was obtained. In a series of joint papers with K. Thomsen coincidence of $E$-theory of Connes–Higson with the functor of $C^*$-algebra extensions was proved. Relations between almost, asymptotic and Fredholm representations of discrete groups were studied. Diagonalization theorem for compact operators on Hilbert $C^*$-modules was proved for a wide class of $C^*$-algebras.
Main publications:
Manuilov V. M. Diagonalization of compact operators in Hilbert modules over finite $W^*$-algebras // Annals of Global Anal. and Geom. 1995, 13(3), 207–226.
Manuilov V. M., Thomsen K. Quasidiagonal extensions and sequentially trivial asymptotic homomorphisms // Adv. Math. (2000), 154, 258–279.
