## Illinois Journal of Mathematics

### On graded $K$-theory, elliptic operators and the functional calculus

Jody Trout

#### Abstract

Let $A$ be a graded $C^{\ast}$-algebra. We characterize Kasparov's $K$-theory group $\hat{K}_{0}(A)$ in terms of graded $\ast$-homomorphisms by proving a general converse to the functional calculus theorem for self-adjoint regular operators on graded Hilbert modules. An application to the index theory of elliptic differential operators on smooth closed manifolds and asymptotic morphisms is discussed.

#### Article information

Source
Illinois J. Math., Volume 44, Issue 2 (2000), 294-309.

Dates
First available in Project Euclid: 19 October 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1255984842

Digital Object Identifier
doi:10.1215/ijm/1255984842

Mathematical Reviews number (MathSciNet)
MR1775323

Zentralblatt MATH identifier
0953.19002

#### Citation

Trout, Jody. On graded $K$-theory, elliptic operators and the functional calculus. Illinois J. Math. 44 (2000), no. 2, 294--309. doi:10.1215/ijm/1255984842. https://projecteuclid.org/euclid.ijm/1255984842