##
**Homogeneity of the pure state space of a separable \(C^{\ast}\)-algebra.**
*(English)*
Zbl 1066.46052

Let \(A\) be a C*-algebra; the linear mappings \(b \mapsto xbx\) and \(b \mapsto [a,b] = ab - ba\) are denoted by \(\operatorname{Ad} x\) and \(\operatorname{Ad} a\) respectively, the space of pure states of \(A\) is denoted \(P(A)\). An automorphism \(\alpha\) of \(A\) is called asymptotically inner if there is a continuous family \((u_{t})_{t \in [0,\infty)}\) of unitaries in \(A\) (or \(A + \mathbf{C}\) if \(A\) is non-unital) such that \(\alpha = \lim_{t \to \infty}\) Ad \(u_{t}\). The group of asymptotically inner automophisms of \(A\) is denoted by AInn(\(A\)). The authors’ main theorem is that, for \(A\) separable, \(\omega_{1}, \omega_{2} \in P(A)\) such that \(\operatorname{ker} \pi_{\omega_{1}} = \operatorname{ker} \pi_{\omega_{2}}\), there is an \(\alpha \in \) AInn(\(A\)) such that \(\omega_{1} \alpha = \omega_{2}\). Here \(\mathcal{H}_{\omega},\pi_{\omega}\) and \(\Omega_{\omega}\) denote the Hilbert space, representation and ground-vector associated with a state \(\omega\). The authors find examples of unital non-separable non-nuclear algebras for which the pure-state space is not homogeneous; viz., von Neumann factors of type II\(_{1}\) or type III. They use a cardinality argument to prove inhomogeneity.

The proof of the theorem is an outstanding development of the article of H. Futamura, N. Kataoka and A. Kishimoto [Int. J. Math. 12, No. 7, 813–845 (2001; Zbl 1067.46056)] showing that a separable simple C*-algebra \(A\) is homogeneous under the action of AInn(\(A\)), or in other words that AInn(\(A\)) is transitive on \(A\), which is an obvious consequence of this theorem. Indeed, the article under review follows the reasoning of their proof but the extra arguments needed are intricate. Of the distinctions between the articles one is that the present article weakens equivalence of representations to quasi-equivalence [see J. Dixmier, “Les C*-algèbres et leurs représentations” (Cahiers scientifiques 29,Gauthier–Villars, Paris) (1969; Zbl 0174.18601), §5.2.3] and another is the use of representations with non-compact range. The authors also make extensive use of R. V. Kadison’s transitivity theorem [Proc. Natl. Acad. Sci. USA 43, 273–276 (1957; Zbl 0078.11502)].

For an arbitrary C*-algebra \(A\), let \(\mathcal{F}\) be a finite subset of \(A\) and \(\omega \in P(A)\), \(\pi\) an irreducible representation of \(A\) on a Hilbert space \(\mathcal{H}\), \(E\) a finite-dimensional projection on \(\mathcal{H}\) and \(\varepsilon >0\). The authors prove that \(A\) has the following Property 1.3 (Futamura’s 2.2), viz., there exists an \(x = (x_{1}, \dots x_{n}); x_i \in M_{1,n}(A)\), for some \(n\), such that \(\|xx^{*}\|\leq 1\), \(\pi(xx^{*})E = E\), \(\|\operatorname{ad} a \operatorname{Ad} x\| < \varepsilon\) for all \(a \in \mathcal{F}\). The proof is an extension of work of U. Haagerup [Invent. Math. 74, 305–319 (1983; Zbl 0529.46041)] for injective von Neumann algebras and a proof of the statement of the theorem restricted there to nuclear \(A\) and injective \(\pi(A)''\).

They then show that an arbitrary C*-algebra with Property 1.3 has their Property 1.2, viz., if \(\pi_{\omega}\), for \(\omega \in P(A)\), has non-compact range then there exists a finite set \(\mathcal{G}\) of \(A\) and a \(\delta > 0\) such that, if \(\phi \in P(A)\) with \(\pi_{\phi}\) quasi-equivalent to \(\pi_{\omega}\) and \(|\phi(x) - \omega(x)| < \delta\) for \(x \in \mathcal{G}\), there exists a continuous path of unitaries \((u_{t})_{t \in [0,1]}\) such that \(u(0) = 1, \phi = \omega Au_{1}\) and \(\|\operatorname{Ad} u_{t}(x) -x\| < \varepsilon \) where \(x\) and \(\varepsilon\) are as in property 1.3. To get these unitaries, they construct finite sets of vectors in \(\mathcal{H}_{\omega}\), \(\xi_{i} = \pi(x^{*}_{i} (\Omega))\) and, for a suitable \(\eta\), \(\eta_{i} = \pi(x^{*}_{i} \eta)\), \(i = 1,2,\dots n\), with closely linked Hilbert-space geometries. The proof involves adapting the problem to be able to use Lemma 3.3 of Futamura et al. to construct the unitary operators needed, and this was not easy. Then, by the Lemma, a close enough geometry of the \(\xi_{i}\) and the \(\eta_{i}\) enables construction of a unitary operator \(U\) such that \(\|U(\xi_{i} - \eta_{i})\|\) is small enough.

Now assuming that \(A\) is separable, the argument for the main theorem follows Theorem 2.5 of Futamura et al. (loc. cit.) to prove that Property 1.2 is sufficient for the main theorem to hold.

The proof of the theorem is an outstanding development of the article of H. Futamura, N. Kataoka and A. Kishimoto [Int. J. Math. 12, No. 7, 813–845 (2001; Zbl 1067.46056)] showing that a separable simple C*-algebra \(A\) is homogeneous under the action of AInn(\(A\)), or in other words that AInn(\(A\)) is transitive on \(A\), which is an obvious consequence of this theorem. Indeed, the article under review follows the reasoning of their proof but the extra arguments needed are intricate. Of the distinctions between the articles one is that the present article weakens equivalence of representations to quasi-equivalence [see J. Dixmier, “Les C*-algèbres et leurs représentations” (Cahiers scientifiques 29,Gauthier–Villars, Paris) (1969; Zbl 0174.18601), §5.2.3] and another is the use of representations with non-compact range. The authors also make extensive use of R. V. Kadison’s transitivity theorem [Proc. Natl. Acad. Sci. USA 43, 273–276 (1957; Zbl 0078.11502)].

For an arbitrary C*-algebra \(A\), let \(\mathcal{F}\) be a finite subset of \(A\) and \(\omega \in P(A)\), \(\pi\) an irreducible representation of \(A\) on a Hilbert space \(\mathcal{H}\), \(E\) a finite-dimensional projection on \(\mathcal{H}\) and \(\varepsilon >0\). The authors prove that \(A\) has the following Property 1.3 (Futamura’s 2.2), viz., there exists an \(x = (x_{1}, \dots x_{n}); x_i \in M_{1,n}(A)\), for some \(n\), such that \(\|xx^{*}\|\leq 1\), \(\pi(xx^{*})E = E\), \(\|\operatorname{ad} a \operatorname{Ad} x\| < \varepsilon\) for all \(a \in \mathcal{F}\). The proof is an extension of work of U. Haagerup [Invent. Math. 74, 305–319 (1983; Zbl 0529.46041)] for injective von Neumann algebras and a proof of the statement of the theorem restricted there to nuclear \(A\) and injective \(\pi(A)''\).

They then show that an arbitrary C*-algebra with Property 1.3 has their Property 1.2, viz., if \(\pi_{\omega}\), for \(\omega \in P(A)\), has non-compact range then there exists a finite set \(\mathcal{G}\) of \(A\) and a \(\delta > 0\) such that, if \(\phi \in P(A)\) with \(\pi_{\phi}\) quasi-equivalent to \(\pi_{\omega}\) and \(|\phi(x) - \omega(x)| < \delta\) for \(x \in \mathcal{G}\), there exists a continuous path of unitaries \((u_{t})_{t \in [0,1]}\) such that \(u(0) = 1, \phi = \omega Au_{1}\) and \(\|\operatorname{Ad} u_{t}(x) -x\| < \varepsilon \) where \(x\) and \(\varepsilon\) are as in property 1.3. To get these unitaries, they construct finite sets of vectors in \(\mathcal{H}_{\omega}\), \(\xi_{i} = \pi(x^{*}_{i} (\Omega))\) and, for a suitable \(\eta\), \(\eta_{i} = \pi(x^{*}_{i} \eta)\), \(i = 1,2,\dots n\), with closely linked Hilbert-space geometries. The proof involves adapting the problem to be able to use Lemma 3.3 of Futamura et al. to construct the unitary operators needed, and this was not easy. Then, by the Lemma, a close enough geometry of the \(\xi_{i}\) and the \(\eta_{i}\) enables construction of a unitary operator \(U\) such that \(\|U(\xi_{i} - \eta_{i})\|\) is small enough.

Now assuming that \(A\) is separable, the argument for the main theorem follows Theorem 2.5 of Futamura et al. (loc. cit.) to prove that Property 1.2 is sufficient for the main theorem to hold.

Reviewer: Aubrey Wulfsohn (Coventry)

### MSC:

46L40 | Automorphisms of selfadjoint operator algebras |

46L30 | States of selfadjoint operator algebras |