New York Journal of Mathematics

New York J. Math. 12 (2006) 269–273.

Real structure in unital separable simple C^{∗}-algebras with tracial rank zero and with a

unique tracial state

P. J. Stacey

Abstract. Let A be a simple unital C^{∗}-algebra with tracial rank zero and with a unique tracial state and let Φ be an involutory ∗-antiautomorphism of A. It is shown that the associated real algebra A_{Φ} = {a ∈ A : Φ(a) = a^{∗}} also has tracial rank zero.

Let A be a unital simple separable C^{∗}-algebra with tracial rank zero and suppose that A has a unique tracial state. If Φ is an involutory ∗-antiautomorphism of A, then it is clear that the associated real algebra A_{Φ} = {a ∈ A : Φ(a) = a^{∗}} is unital and simple with a unique tracial state, but it is not clear that it has tracial rank zero, even when A is approximately finite-dimensional.

The purpose of the present note is to show that techniques recently developed by Phillips [14] and Osaka and Phillips [12], [13] can be used to show that A_{Φ} does have tracial rank zero. This raises the possibility of classifying all real structures in the algebras under consideration by developing a real analogue of Lin’s classification [10] of C^{∗}-algebras of tracial rank zero. Previously all classifications of real structures in non-type I simple C^{∗}-algebras, such as [2], [3], [15] for AF algebras and [5], [16] for irrational rotation algebras, have assumed very specific forms for the real algebras.

The key step in showing that A_{Φ} has tracial rank zero is to show that Φ has the tracial Rokhlin property, defined below, as introduced in Definition 1.1 of [14].

Definition 1. Let A be a stably finite simple unital C^{∗}-algebra and let Φ be an involutory ∗-antiautomorphism of A. Then Φ has the tracial Rokhlin property if for every finite set F ⊂ A, every ϵ > 0, every N ∈ ℕ and every nonzero positive element x ∈ A there are mutually orthogonal projections e_{0}, e_{1} ∈ A such that:

(1) ∥Φ(e_{0}) − e_{1}∥ < ϵ.

(2) ∥e_{j}a − ae_{j}∥ < ϵ for 0 ≤ j ≤ 1 and all a ∈ F.

(3) The projection 1−e_{0}−e_{1} is Murray–von Neumann equivalent to a projection in the hereditary subalgebra of A generated by x.

(4) For every 0 ≤ j ≤ 1 there are N mutually orthogonal projections

f_{1}, …, f_{N} ≤ e_{j},

269^{ISSN 1076-9803/06}

Received September 8, 2005.

Mathematics Subject Classification. 46L35,46L40,46L05.

Key words and phrases. Real C^{∗}-algebras, tracial rank, tracial Rokhlin property, involutory antiautomorphism.

each of which is Murray–von Neumann equivalent to the projection 1−e_{0}−e_{1}.

Remark 2. As in Lemma 1.4 of [12], with ϵ chosen sufficiently small to ensure (4), if A has real rank zero and the order on projections is determined by traces then conditions (3) and (4) in Definition 1 can be replaced by:

(3′) τ(1 − e_{0} − e_{1}) < ϵ for all tracial states τ.

Following Theorem 2.14 of [13], a further reformulation of the definition can be given in terms of the L^{2}-seminorm associated with a tracial state τ, defined by ∥a∥_{2,τ} = τ(a^{∗}a)^{1/2}.

Lemma 3. Let A be a simple separable unital C^{∗}-algebra with tracial rank zero, let Φ be an involutory ∗-antiautomorphism of A and let T (A) be the tracial state space of A. Then Φ has the tracial Rokhlin property if and only if for every ϵ > 0 and every finite subset S of A there exists a projection e ∈ A such that:

(1) ∥Φ(e) + e − 1∥_{2,τ} < ϵ for all τ ∈ T (A).

(2) ∥[e, a]∥_{2,τ} < ϵ for all a ∈ S and all τ ∈ T (A).

Proof. The proof directly follows that of Theorem 2.14 in [13], although the present situation is considerably simpler. As noted there, A satisfies the conditions of Remark 2. Thus, if Φ satisfies the tracial Rokhlin property, there exist e_{0}, e_{1} with ∥Φ(e_{0}) − e_{1}∥ < \frac{1}{2}ϵ, τ(1 − e_{0} − e_{1}) < \frac{1}{4}ϵ^{2} and ∥[e_{0}, a]∥ < ϵ for each a ∈ S. For τ ∈ T (A) and a ∈ S it follows that ∥[e_{0}, a]∥_{2,τ} < ϵ and

∥Φ(e_{0}) + e_{0} − 1∥_{2,τ} ≤ ∥Φ(e_{0}) − e_{1}∥_{2,τ} + ∥1 − e_{0} − e_{1}∥_{2,τ}

< \frac{1}{2}ϵ + τ(1 − e_{0} − e_{1})^{1/2} < ϵ.

The converse holds by the argument in Theorem 2.14 in [13], applied to the real-linear automorphism α = Φ ∘ ∗ (and n = 1): the Lemma 2.13 used in the proof holds equally well for a real linear action. □

The following result now follows as in Theorem 2.17 of [13] , using a property of involutory ∗-antiautomorphisms of continuous von Neumann algebras from [1].

Theorem 4. Let A be a simple separable unital C^{∗}-algebra with tracial rank zero and with a unique tracial state τ. Then any involutory ∗-antiautomorphism Φ of A has the tracial Rokhlin property.

Proof. The conditions of Lemma 3 will be demonstrated, so let ϵ > 0 and let S be a finite subset of A with ∥a∥ ≤ 1 for each a ∈ S. If τ is the unique tracial state, let N = π_{τ} (A)′′ and, for ω ∈ βℕ\ℕ, let N_{ω} be the central sequence algebra and let Φ_{ω} be the involutory antiautomorphism of N_{ω} arising from Φ. N_{ω} is a continuous von Neumann algebra (being a type II_{1} factor, as observed in the proof of Theorem 2.17 of [13]) so, by Lemme 1.8 of [1], there exists a 2 × 2 set of matrix units {e_{i,j}}_{1≤i,j≤2} with Φ_{ω}(e_{i,j}) = e_{j,i}. Then f = \frac{1}{2}(1+ie_{1,2}−ie_{2,1}) is a projection with Φ_{ω}(f) = 1−f. As in the proof of Theorem 2.17 of [13], represent f by a sequence (f_{ℓ})_{ℓ∈ℕ} in ℓ^{∞}(N) such that each f_{ℓ} is a projection and let U be a neighbourhood of ω in βℕ such that ℓ ∈ U implies ∥[a, f_{ℓ}]∥_{2,τ} < \frac{1}{3}ϵ for a ∈ S. Let ℓ_{0} ∈ ℕ satisfy ℓ_{0} ∈ U and ∥\overline{Φ}(f_{ℓ_{0}}) − (1 − f_{ℓ_{0}})∥_{2,τ} < \frac{1}{3}ϵ, where \overline{Φ} is the extension of Φ to N. By Lemma 2.15 of [13] there exists a projection e ∈ A with ∥e − f_{ℓ_{0}}∥_{2,τ} < \frac{1}{3}ϵ and therefore

∥e + Φ(e) − 1∥_{2,τ} ≤ ∥e − f_{ℓ_{0}}∥_{2,τ} + ∥f_{ℓ_{0}} − \overline{Φ}(1 − f_{ℓ_{0}})∥_{2,τ}

+ ∥\overline{Φ}(1 − f_{ℓ_{0}}) − Φ(1 − e)∥_{2,τ}

< \frac{1}{3}ϵ + \frac{1}{3}ϵ + \frac{1}{3}ϵ = ϵ.

Also

∥[e, a]∥_{2,τ} ≤ 2∥e − f_{ℓ_{0}}∥_{2,τ} ∥a∥ + ∥[f_{ℓ_{0}}, a]∥_{2,τ} < \frac{2}{3}ϵ + \frac{1}{3}ϵ = ϵ.  □

Theorem 4 will be applied as in Theorem 2.7 of [14], but this invokes a result of Jeong and Osaka from [6] which in turn invokes a result of Kishimoto [7] on outer automorphisms. An analogue of Kishimoto’s result for antiautomorphisms has been obtained in [4], but this result is not quite strong enough to establish the analogue of Jeong and Osaka’s result. However Tomohiro Hayashi has informed the author of the following strengthening of his result from [4] which serves the required purpose.

Theorem 5 (Hayashi). Let A be a non-type I separable simple unital C^{∗}-algebra and let α be an antiautomorphism on A. Then, for any hereditary C^{∗}-subalgebra B ⊂ A and for any a ∈ A, we have

inf {||xaα(x)|| : x ∈ B^{+}, ||x|| = 1} = 0.

Proof. For the proof, it is enough to show that for any unitaries u_{1}, ⋯, u_{n} ∈ A, any hereditary C^{∗}-subalgebra B ⊂ A and any positive number ϵ > 0, we can find an element x ∈ B^{+} such that ||x|| = 1 and ||xu_{i}α(x)|| < ϵ (i = 1, ⋯, n). (The element a can be expressed as a linear combination of unitaries.) By Theorem 2.1 of [4] we can find an element x_{1} ∈ B^{+} such that ||x_{1}|| = 1 and ||x_{1}u_{1}α(x_{1})|| < ϵ since x ↦ u_{1}α(x)u_{1}^{∗} is an antiautomorphism. Moreover, replacing x_{1} by a suitable function f(x_{1}) if necessary, we may assume that there exists a positive element c_{1} ∈ B such that ||c_{1}|| = 1 and c_{1}x_{1} = x_{1}c_{1} = c_{1}. Then applying Theorem 2.1 of

[4] again, we can find an element x_{2} ∈ \overline{(c_{1}Ac_{1})}^{||·||}(⊂ B) such that ||x_{2}|| = 1 and ||x_{2}u_{2}α(x_{2})|| < ϵ. Here we remark that

||x_{2}u_{1}α(x_{2})|| = ||x_{2}x_{1}u_{1}α(x_{2}x_{1})|| ≤ ||x_{1}u_{1}α(x_{1})|| < ϵ

because of the choice of c_{1}. Therefore, by induction we get the desired element x = x_{n}. □

The relevant consequence of this result is the following simple variant of Theorem 4.2 of [6].

Lemma 6. Let A be a simple unital C^{∗}-algebra in which every nonzero hereditary C^{∗}-subalgebra has a nonzero projection, let Φ be an involutory ∗-antiautomorphism of A and let α = Φ ∘ ∗. Then any nonzero hereditary C^{∗}-subalgebra of the crossed product A ×_{α} ℤ_{2} contains a nonzero projection which is equivalent to a projection in A.

Proof. As in the proof of Theorem 4.2 of [6], let a be a positive element of norm 1 in A ×_{α} ℤ_{2} and write a = a_{0} + a_{1}δ_{1} where δ_{1} is the unitary implementing α. Note that Theorem 5 enables Lemma 3.2 of [7] to be applied, given ϵ > 0, to produce a positive element x ∈ A of norm 1 with ∥xa_{0}x∥ > (1 − ϵ)||a_{0}|| and ||xa_{1}α(x)|| < ϵ/4 and hence with ||xa_{1}δ_{1}x|| < ϵ/4. The proof of Theorem 4.2 of [6] then applies directly (with N = {1} and a = b^{∗}b) to produce the required projection. □

The required analogue of Theorem 2.7 of [14] can now be obtained. The criterion used here for a simple real C^{∗}-algebra to have tracial rank zero is the real analogue of Proposition 2.1 of [14], as follows:

Definition 7. A simple separable unital real C^{∗}-algebra is said to have tracial topological rank zero if the following holds. For every finite subset S of A, every ϵ > 0, every nonzero positive x ∈ A and every N ∈ ℕ, there exists a projection p ∈ A and a finite-dimensional unital subalgebra E of pAp such that:

(1) ∥pa − ap∥ < ϵ for all a ∈ S.

(2) For every a ∈ S there exists b ∈ E such that ∥pap − b∥ < ϵ.

(3) 1 − p is Murray–von Neumann equivalent to a projection in \overline{xAx}.

(4) There are N mutually orthogonal projections in pAp, each Murray–von Neumann equivalent to 1 − p.

Remark 8. As in Proposition 2.1 of [14], it can be shown that if in addition a_{0} ∈ A is a given nonzero element, then E and p can be chosen so that ||pa_{0}p|| < ||a_{0}|| − ϵ.

Theorem 9. Let A be a simple unital C^{∗}-algebra with tracial rank zero and let Φ be an involutory ∗-antiautomorphism of A with the tracial Rokhlin property. Then the associated real algebra A_{Φ} = {a ∈ A : Φ(a) = a^{∗}} has tracial rank zero.

Proof. Recall firstly that, with α = Φ ∘ ∗, A ×_{α} ℤ_{2} is isomorphic to the algebra M_{2}(A_{Φ}) of 2 × 2 matrices over A_{Φ}, with the element a + ib of A = A_{Φ} + iA_{Φ} corresponding to the element a(e_{11}+e_{22})+b(e_{12}−e_{21}) of M_{2}(A_{Φ}) and the canonical unitary to e_{12} + e_{21}. If it can be shown that A ×_{α} ℤ_{2} has tracial rank zero then, as in Lemma 3.6.5 of [11], it follows that e_{11}M_{2}(A_{Φ})e_{11} has tracial rank zero, giving the required result.

A considerably simpler version of the argument in Theorem 2.7 of [14], applied to α = Φ ∘ ∗, shows that A ×_{α} ℤ_{2} does indeed have tracial rank zero. The only result quoted in that proof which does not immediately carry through to the current context is Theorem 4.2 of [6], which is replaced by Lemma 6 above. □

Corollary 10. Let A be a simple unital C^{∗}-algebra with tracial rank zero and with a unique tracial state and let Φ be an involutory ∗-antiautomorphism of A. Then the associated real algebra A_{Φ} = {a ∈ A : Φ(a) = a^{∗}} has tracial rank zero.

Proof. This is immediate from Theorems 4 and 9. □

References

[1] Giordano, T. Antiautomorphismes involutifs des facteurs de von Neumann injectifs. I. J. Operator Theory 10 (1983), 251–287. MR0728909 (85h:46087), Zbl 0533.46039.

[2] Giordano, T. A classification of approximately finite real C^{∗}-algebras. J. Reine Angew Math. 385 (1988), 161–194. MR0931219 (89h:46078), Zbl 0635.46055.

[3] Goodearl, K. R.; Handelman, D. E. Classification of ring and C^{∗}-algebra direct limits of finite-dimensional semisimple real algebras. Mem. Amer. Math. Soc. 69 (1987), no. 372. MR0904013 (88k:46067), Zbl 0629.46053.

[4] Hayashi, Tomohiro. A Kishimoto type theorem for antiautomorphisms with some applications. Internat. J. Math. 15 (2004), 487–499. MR2072091 (2005f:46117), Zbl 1056.46055.

[5] Hu, Yaohua; Stacey, P. J. Toral automorphisms and antiautomorphisms of rotation algebras. Bull. Aust. Math. Soc. 59 (1999), 247–255. MR1680795 (2000e:46078), Zbl 0956.46042.

Real tracial rank zero 273

[6] Jeong, Ja A; Osaka, Hiroyuki. Extremally rich C^{∗}-crossed products and the cancellation property. J. Austral. Math. Soc. (Series A) 64 (1998), 285–301. MR1623278 (99k:46097), Zbl 0919.46041.

[7] Kishimoto, Akitaka. Outer automorphisms and reduced crossed products of simple C^{∗}-algebras. Commun. Math. Phys. 81 (1981), 429–435. MR0634163 (83c:46061), Zbl 0467.46050.

[8] Lin, Huaxin. Tracially AF C^{∗}-algebras. Trans. Amer. Math. Soc. 353 (2001), 693–722. MR1804513 (2001j:46089), Zbl 0964.46044.

[9] Lin, Huaxin. The tracial topological rank of C^{∗}-algebras. Proc. London Math. Soc. 83 (2001), 199–234. MR1829565 (2002e:46063), Zbl 1015.46031.

[10] Lin, Huaxin. Classification of simple C^{∗}-algebras of tracial topological rank zero. Duke Math. J. 125 (2004), 91–119. MR2097358 (2005i:46064).

[11] Lin, Huaxin. An introduction to the classification of amenable C^{∗}-algebras. World Scientific, River Edge, NJ, 2001. MR1884366 (2002k:46141), Zbl 1013.46055.

[12] Osaka, Hiroyuki; Phillips, N. Christopher. Stable and real rank for crossed products by automorphisms with the tracial Rokhlin property. Preprint math.OA/0409168.

[13] Osaka, Hiroyuki; Phillips, N. Christopher. Furstenberg transformations on irrational rotation algebras. Preprint math.OA/0409169.

[14] Phillips, N. Christopher. Crossed products by finite cyclic group actions with the tracial Rokhlin property. Preprint math.OA/0306410.

[15] Stacey, P.J. Real structure in direct limits of finite-dimensional C^{∗}-algebras. J. London Math. Soc. (2) 35 (1987), 339–352. MR0881522 (88g:46083), Zbl 0591.46049.

[16] Stacey, P.J. Inductive limit decompositions of real structures in irrational rotation algebras. Indiana Univ. Math. J. 51 (2002), 1511–1540. MR1948458 (2004e:46077), Zbl 1032.46071.

Department of Mathematics, La Trobe University, Victoria 3086, Australia P.Stacey@latrobe.edu.au

This paper is available via http://nyjm.albany.edu/j/2006/12-17.html.