Pre-C*-algebras

If any spectral triple $(\sA, \sH, D)$, the algebra $\sA$ is a (unital) $*$-algebra of bounded operators acting on a Hilbert space $\sH$ [or, if one wishes to regard $\sA$ abstractly, a faithful representation $\pi \: \sA \to \sL(\sH)$ is given]. Let $A$ be the norm closure of $\sA$ [or of $\pi(\sA)$] in $\sL(\sH)$: it is a $C^*$-algebra in which $\sA$ is a dense $*$-subalgebra.

A priori, the only functional calculus available for $\sA$ is the holomorphic one:

$$f(a) := \inv{2\pi i} \oint_\Ga f(\la) (\la 1 - a)^{-1} d\la,$$ (25)

where $\Ga$ is a contour in $\bC$ winding (once positively) around $\spec(a)$, and $\spec(a)$ means the spectrum of $a$ in the $C^*$-algebra $A$. To ensure that $a \in \sA$ implies $f(a) \in \sA$, we need the following property:

If $a \in \sA$ has an inverse $a^{-1} \in A$, then in fact $a^{-1}$ lies in $\sA$ (briefly: $\sA \cap A^\x = \sA^\x$, where $\sA^\x$ is the group of invertible elements of $A$). If this condition holds, then $\inv{2\pi i} \oint_\Ga f(\la) (\la 1 - a)^{-1} d\la$ is a limit of Riemann sums lying in $\sA$. To ensure convergence in $\sA$ (they do converge in $A$), we need only ask that $\sA$ be complete in some topology that is finer than the $C^*$-norm topology.

If $\sA$ is a nonunital algebra, we can always adjoin a unit in the usual way, and work with $\Tilde\sA := \bC \oplus \sA$ whose unit is $(1,0)$, and with its $C^*$-completion $\Tilde A := \bC \oplus A$. Since the multiplication rule in $\Tilde\sA$ is $(\la,a)(\mu,b) := (\la\mu, \mu a + \la b + ab)$, we see that $1 + a := (1,a)$ is invertible in $\Tilde\sA$, with inverse $(1,b)$, if and only if $a + b + ab = 0$.

We state, without proof, two important facts about Fréchet pre-$C^*$-algebras.

This invariance of $\rK$-theory was proved by Bost [Bost]. For $\rK_0$, the spectral invariance plays the main role. For $\rK_1$, one must first formulate a topological $\rK_1$-theory is a category of ``good'' locally convex algebras (thus whose invertible elements form an open subset and for which inversion is continuous), and it is known that Fréchet pre-$C^*$-algebras are ``good'' in this sense.

These properties of the completed spectral triple are due to Rennie [Ren]. We now discuss another result of Rennie, namely that such completed algebras of regular spectral triples are endowed with a $\Coo$ functional calculus.

Before showing how this smooth functional calculus can yield useful results, we pause for a couple of technical lemmas on approximation of idempotents and projectors, in Fréchet pre-$C^*$-algebras. The first is an adaptation of a proposition of [Bost].

Lemma  says that in a unital Fréchet pre-$C^*$-algebra $\sA$, an ``almost idempotent'' $v \in \sA$ that is not far from being a projector (since $\Vert 1 - 2v\Vert$ is close to $1$) can be retracted to a genuine idempotent in $\sA$. The next Lemma says that projectors in the $C^*$-completion of $\sA$ can be approximated by projectors lying in $\sA$.