I'm currently starting to learn K-homology and something bothers me about the definition of Fredholm modules. I looked in 5/6 papers and each time a different definition is given... For example in [1] and [2]:
Definition (Fredholm module) : Let $\mathcal{A}$ be a separable $C^*$-algebra. A Fredholm module over $\mathcal{A}$ is given by a triple $(\mathcal{H},\rho,F)$ where :
- a Hilbert space $\mathcal{H}$
- a $*$-representation $\rho : \mathcal{A} \rightarrow \mathcal{B}(\mathcal{H})$
- A operator $F : \mathcal{H} \rightarrow \mathcal{H}$ such that for all $a \in \mathcal{A}$, $$[F, \rho(a)],\qquad (F^2 - 1)\rho(a),\qquad (F^* - F)\rho(a) $$ are in $\mathcal{K}(\mathcal{H})$ the ideal of compact operators.
In a lot of other references (for example [3],[4]...), they give the same definition but in addition they require that $F$ be bounded on $\mathcal{H}$ .
Then,
Questions :
Are the 2 definitions equivalent? (maybe the conditions on $F$ with the compact perturbations imply that $F$ is bounded but I don't manage to prove it). If not, can you give me an operator which is not bounded and verify the definition?
(In some papers they also define bounded (resp. unbounded) Fredholm module and in this case the operators in the definition are bounded (resp. unbounded) but I would like to know if a Fredholm module (without the adjective bounded or unbounded ) refers to one of the two cases or not))
[1] A. Connes, Noncommutative Geometry, Academic Press, 1994.
[2] Nigel Higson and John Roe, Analytic K-homology.
[3] G. Kasparov. Topological invariants of elliptic operators, I. K-homology
[4] A. L. Carey, J. Phillips, F.A. Sukochev, On unbounded p-summable Fredholm module