Bounded commutator - Imported Knowlpedia Content

Definition

Let $E$ be a dense subspace of a Hilbert space $H$ , and suppose the operator commutator $[A,B]$ is defined on $E$ . It is a bounded commutator if there is a constant $C\geq 0$ such that

\|[A,B]\xi\|\leq C\|\xi\|\qquad(\xi\in E).

The commutator then extends uniquely by continuity to a bounded operator on $H$ . This extension, rather than the generally unbounded products $AB$ and $BA$ , is also denoted $[A,B]$ . Density of $E$ is essential for uniqueness; an estimate on a nondense common domain does not determine an operator on all of $H$ .

Extension criterion

The defining estimate makes $[A,B]:E\to H$ continuous for the norm inherited from $H$ . Completing $E$ therefore produces the unique bounded extension, whose norm is

\|[A,B]\|=\sup_{\xi\in E,\ \xi\neq0} \frac{\|[A,B]\xi\|}{\|\xi\|}.

No separate assumption that the initial commutator is closed is needed. Its graph closure is precisely the graph of the bounded extension restricted to the closure of $E$ .

Spectral-triple convention

In a spectral triple $(\mathcal A,H,D)$ , the standard axiom means that every represented element $\pi(a)$ preserves $\operatorname{Dom}(D)$ and that the commutator $[D,\pi(a)]$ , initially on $\operatorname{Dom}(D)$ , satisfies the bounded estimate above. The same symbol is then used for its unique extension to $H$ . Requiring only a bounded form commutator, or defining the expression on a smaller core, gives a variant of the axiom and should not be silently identified with this convention Connes and Moscovici, opening spectral-triple convention.

Examples and scope

For bounded operators $A$ and $B$ , the commutator is automatically bounded and satisfies $\|[A,B]\|\leq2\|A\|\|B\|$ . More geometrically, for $D=-i\,d/dx$ and multiplication $M_f$ on the circle, the identity $[D,M_f]=-iM_{f'}$ shows that the commutator is bounded when $f'$ is essentially bounded. The axiom controls one derivative of $f$ ; it does not assert that either product $DM_f$ or $M_fD$ is bounded.

References

Alain Connes and Henri Moscovici, “The Local Index Formula in Noncommutative Geometry,” Geometric and Functional Analysis 5 (1995), 174–243. DOI record. Relevant: the spectral-triple hypotheses at the beginning of the article.
José M. Gracia-Bondía, Joseph C. Várilly, and Héctor Figueroa, Elements of Noncommutative Geometry, Birkhäuser, 2001. DOI record. Relevant: §10.1 on the domain and bounded-extension convention.