CUBO A Mathematical Journal Vol.10, N o ¯ 02, (75–82). July 2008 Spectral Rank for C∗-Algebras Takahiro Sudo Department of Mathematical Sciences, Faculty of Science, University of the Ryukyus, Nishihara, Okinawa 903-0213, Japan email: sudo@math.u-ryukyu.ac.jp ABSTRACT We introduce a notion of dimension for C∗-algebras that we call spectral rank, based on spectrums of generators of C∗-algebras. We study some basic properties for this new rank and establish its fundamental theory. RESUMEN Introducimos la noción de dimensión para C∗-algebras que llamamos rango espectral; esta noción es basada en el espectro de los generadores de C∗-algebras. Estudiamos algunas propriedades básicas para este nuevo rango y establecemos su teoria funda- mental. Key words and phrases: C*-algebra, Rank, Spectrum. Math. Subj. Class.: 46L05 76 Takahiro Sudo CUBO 10, 2 (2008) Introduction There have been several attempts to introduce suitable ranks for C∗-algebras; the stable rank (and connected stable rank) of Rieffel [4], the real rank of Brown and Pedersen [1], the completely positive (or decomposition) rank (or covering dimension) of Winter [7], and the topological rank and (another) covering dimension of the author [5], [6], etc. For reference, see [2] or [3]. In this paper we introduce a yet another notion of dimension for C∗-algebras that we call spectral rank. This rank is based on spectrums of generators of C∗-algebras. We study some basic properties for this new rank that might become an interesting new invariant for C∗-algebras. In Section 1, some basic properties of the spectral rank for C∗-algebras concerning their fundamental algebraic structures are discussed. In Section 2, introduced is an approximate version of the spectral rank that we call approximate spectral rank. 1 Spectral rank Let A be a C∗-algebra. The spectrum of an element a of A is defined by sp(a) = {λ ∈ C | a − λ1 is not invertible in A + }, where A + = A when A is unital, and A + is the unitization of A by C of complex numbers when A is non-unital. Note that the spectrum sp(a) is a non-empty closed subset of C and bounded by the norm ‖a‖. Definition 1.1. Let A be a C∗-algebra with (specific and initial) generators aj . Define the spectral rank of A to be spr(A) = inf    ∑ j dim sp(aj ) | aj are generators of A    where dim(·) is the (covering) dimension for spaces. Remark. This notion should not depend on generators but do depend in a sense, and certainly does not depend on their certain equivalences like adjoint unitary operations since sp(aj ) = sp(Ad(u)aj ) where Ad(u)aj = uaju ∗ for some unitary u, and some cancellative terms. In another view, we just look at generators of the algebraic part of A and ignore some unknown ones in the C∗-closure of the part in A, and always consider such a situation in what follows. Proposition 1.2. Let A be a C∗-algebra with unitary generators uj for j ∈ J a set. Then spr(A) ≤ ∑ j dim sp(uj ) ≤ ∑ j 1 = |J|, where the second inequality is equality if A is universal. CUBO 10, 2 (2008) Spectral Rank for C∗-Algebras 77 Proof. Note that sp(uj ) is a closed subset of the torus T with dim T = 1. Thus, dim sp(uj) ≤ 1. If A is universal, we have sp(uj ) = T. 2 Proposition 1.3. Let A be a C∗-algebra and B its quotient C∗-algebra, where generators of B are mapped from those of A by the quotient map. Then spr(A) ≥ spr(B). Proof. Let π be the quotient map from A to B. Let aj be generators of A. Then π(aj ) are generators of B and note that sp(π(aj )) ⊂ sp(aj ). 2 Example 1.4. Let C(Tn) be the C∗-algebra of all continuous functions on the n-torus Tn. This is the universal C∗-algebra generated by mutually commuting n unitaries. Hence, spr(C(Tn)) = n. Let T n θ be the noncommutative n-torus, which is defined to be the universal C∗-algebra gen- erated by n unitaries uj such that ujui = e 2πiθij uiuj for 1 ≤ i, j ≤ n, where θ = (θij ) is a skew-adjoint n × n matrix over R. Thus, spr(Tn θ ) = n. Let C∗(Fn) be the full group C ∗ -algebra of the free group Fn with n generators. This is the universal C∗-algebra generated by n unitaries with no relations. Hence, spr(C∗(Fn)) = n. Proposition 1.5. Let A be a C∗-algebra with isometry generators sj for j ∈ J a set. Then spr(A) ≤ ∑ j dim sp(sj ) = ∑ j 2 = 2|J|, and the inequality spr(A) ≤ 2|J| holds in general, with {aj}j∈J generators of A. Proof. Note that sp(sj ) is the unit disk D with dim D = 2. Thus, dim sp(sj ) = 2. In general, sp(aj ) is a closed subset of C, so that dim sp(aj ) ≤ 2. 2 Example 1.6. Let F be the Toeplitz algebra, which is the universal C∗-algebra generated by an isometry. Thus, spr(F) = 2. Let C∗(Nn) be the full semigroup C ∗ -algebra of the free semigroup Nn with n generators, i.e., Nn ∼= ∗ n N the n-fold free product of the semigroup N of natural numbers. This is the universal C∗-algebra generated by n isometries with no relations. Hence, spr(C∗(Nn)) = 2n. Let On be the Cuntz algebra (2 ≤ n < ∞), which is the universal C ∗ -algebra generated by n isometries sj such that ∑ n j=1 sj s ∗ j = 1. Then spr(On) = 2n. Let O∞ be the Cuntz algebra generated by isometries sj (j ∈ N) such that ∑ n j=1 sj s ∗ j < 1. Then spr(O∞) = ∞. Proposition 1.7. Let A be a C∗-algebra generated by compact operators. Then spr(A) = 0. 78 Takahiro Sudo CUBO 10, 2 (2008) Proof. Note that for a compact opearator T , we have dim sp(T ) = 0. 2 Proposition 1.8. Let A, B be C∗-algebras and A ⊕ B their direct sum. Then spr(A ⊕ B) = spr(A) + spr(B). Proof. Suppose that aj are generators of A and bj are those of B. Then aj ⊕ 0 and 0 ⊕ bj are generators of A ⊕ B. 2 Example 1.9. For Mn(C) the n × n matrix algebra over C, spr(Mn(C)) = 0. Let K be the C∗-algebra of all compact operators on a separable infinite-dimensional Hilbert space. Then spr(K) = 0. An AF algebra that is an inductive limit of finite dimensional C∗-algebras (i.e., finite direct sums of some Mn(C)) has spectral rank zero. Proposition 1.10. Let A be a C∗-algebra and I its C∗-subalgebra, where generators of I can be always taken from those of A. Then spr(I) ≤ spr(A). In particular, we may take I as a closed two-sided ideal or hereditary C∗-subalgebra in this sense. Proof. Generators of I can be viewed as a part of those of A. 2 Remark. The assumption for generators is necessary. Indeed, there exist some C∗-algebras that are embeddable into AF algebras, but they should have spectral rank non-zero, such as rotation C∗-algebras. This is an obstruction to our theory, but it seems that in this case the generators of those C∗-algebras are not to be visible in AF. 2 Proposition 1.11. Let 0 → I → A → A/I → 0 be a short exact sequence of C∗-algebras, where generators of I are always taken from those of A. Then spr(A) ≥ max{spr(I), spr(A/I)}. Remark. It is likely but not true in general that spr(A) ≤ spr(I) + spr(A/I). For instance, the Toeplitz algebra F = C∗(S) is decomposed into 0 → K → F → C(T) → 0, where K is generated by some elements like the finite rank projections 1 − Sn(Sn)∗ for n ∈ N and S is mapped to the generator of C(T). But spr(F) = 2, spr(K) = 0, and spr(C(T)) = 1. However, the generators of K are not a part of those of F. CUBO 10, 2 (2008) Spectral Rank for C∗-Algebras 79 Proposition 1.12. Let A be a C∗-algebra and A+ its unitization by C. Then spr(A) = spr(A + ). Proof. Note that for the unit 1, sp(1) = {1} ⊂ C. Thus, dim sp(1) = 0. 2 Proposition 1.13. Let 0 → I → A → B → 0 be an extension of C∗-algebras. Then spr(A) ≤ max{spr(M (I)), spr(B)}. where M (I) is the multiplier algebra of I, and generators of A are viewed as part of those of the direct sum M (I) ⊕ B containing the pullback C∗-algebra associated with the extension. Proof. It is well known that A is isomorphic to the pullback C∗-algebra in M (I) ⊕ B with the associated Busby map from B to M (I)/I and the quotient map from M (I) to M (I)/I. 2 Proposition 1.14. Let A, B be C∗-algebras and A ⊗ B their C∗-tensor product with a C∗-norm. Then max{spr(A), spr(B)} ≤ spr(A ⊗ B) ≤ spr(A) + spr(B). Proof. The left inequality is clear since A, B are C∗-subalgebras of A ⊗ B preserving generators. Assume first that A, B are unital. Suppose that aj are generators of A and bj are those of B. Then aj ⊗ 1 and 1 ⊗ bj are generators of A ⊗ B. If A or B are non-unital, then spr(A ⊗ B) ≤ spr(A + ⊗ B + ) ≤ spr(A + ) + spr(B + ) = spr(A) + spr(B) since A ⊗ B is a closed ideal of A+ ⊗ B+. 2 Corollary 1.15. For Mn(A) the n × n matrix algebra over a C ∗-algebra A, spr(Mn(A)) = spr(A). Furthermore, if B is a C∗-algebra with spr(B) = 0, then spr(A ⊗ B) = spr(A). Proof. Note that Mn(A) ∼= A ⊗ Mn(C). 2 Proposition 1.16. Let A be a C∗-algebra, G a finitely generated discrete group with n generators, and A ⋊α G a (full or reduced) C ∗-crossed product of A by an action α of G by automorphisms. Then spr(A) ≤ spr(A ⋊α G) ≤ spr(A) + n. In particular, if G = Z, then spr(A) ≤ spr(A ⋊α Z) ≤ spr(A) + 1. 80 Takahiro Sudo CUBO 10, 2 (2008) Proof. The crossed product A ⋊α G is generated by A and the unitaries corresponding to the generators of G (on the universal, or a certain Hilbert space). 2 Corollary 1.17. Let G be a finitely generated discrete group with n generators. Let C∗(G) and C∗ r (G) be its full and reduced group C∗-algebras. Then spr(C∗(G)) ≤ n, and spr(C∗ r (G)) ≤ n. Proposition 1.18. Let A be a C∗-algebra, N a finitely generated discrete semi-group with n generators, and A ⋊β N a (full or reduced) semi-group crossed product of A by an action β of N by endomorphisms. Then spr(A) ≤ spr(A ⋊β N ) ≤ spr(A) + 2n. In particular, if N = N, then spr(A) ≤ spr(A ⋊β N) ≤ spr(A) + 2. Proof. The crossed product A ⋊β N is generated by A and the isometries corresponding to the generators of N (on the universal, or a certain Hilbert space). 2 Corollary 1.19. Let N be a finitely generated discrete semi-group with n generators. Let C∗(N ) and C∗ r (N ) be its full and reduced semi-group C∗-algebras. Then spr(C ∗ (N )) ≤ 2n, and spr(C∗ r (N )) ≤ 2n. Proposition 1.20. Let A be a continuous field C∗-algebra on a locally compact Hausdorff space X with fibers the same C∗-algebra B. Then spr(A) ≤ spr(C0(X)) + spr(B), where C0(X) is the C ∗-algebra of all continuous functions on X vanishing at infinity. If B is unital with spr(B) = 0, then spr(A) = spr(C0(X)). Proof. Assume first that B is unital. Then A is assumed to be generated by generators of C0(X) and those of B. Also, we obtain spr(C0(X)) ≤ spr(A). If B is non-unital, we can consider the unitization A + of A by adding the unit field on X taking the unit of the unitization B + of B. Therefore, we obtain spr(A) = spr(A + ) ≤ spr(C0(X)) + spr(B + ) = spr(C0(X)) + spr(B). 2 CUBO 10, 2 (2008) Spectral Rank for C∗-Algebras 81 Example 1.21. Let T2 θ be the rational rotation C∗-algebra corresponding to a rational θ. This can be viewed as a continuous filed C∗-algebra on the 2-torus T2 with fibers the same Mn(C) with n the period of θ. Thus, spr(T 2 θ ) = spr(C(T2)) = 2. Proposition 1.22. Let A be a C∗-algebra and B a C∗-algebra deformed from A with generators and relations deformed from those of A. Then spr(A) = spr(B). Example 1.23. Let Tn θ be a noncommutative n-torus. This is deformed from C(Tn), and spr(T n θ ) = spr(C(Tn)) = n. Proposition 1.24. Let A, B be C∗-algebras and A∗B their C∗-free product with a (full or reduced) C∗-norm. Then max{spr(A), spr(B)} ≤ spr(A ∗ B) ≤ spr(A) + spr(B). Also, A ∗ B can be replaced with the unital C∗-free product A ∗C B. Proof. The left inequality is clear since A, B are C∗-subalgebras of A ∗ B preserving generators. Suppose that aj are generators of A and bj are those of B. Then aj and bj are generators of A ∗ B. 2 2 Approximate spectral rank Definition 2.1. Let A be a C∗-algebra. Define the approximate spectral rank of A to be the minimum non-negative integer n = aspr(A) such that for any a ∈ A and ε > 0, there exists a C∗-subalgebra B of A with spr(B) ≤ n such that ‖a − b‖ ≤ ε for some b ∈ B. Proposition 2.2. Let A be an inductive limit of C∗-algebras An with spr(An) ≤ sn for some sn. Then aspr(A) ≤ lim sn, where lim is the limit infimum. Example 2.3. If A is an AF-algebra, then aspr(A) = 0 = spr(A). Let A be an AT-algebra, i.e., an inductive limit of finite direct sums of matrix algebras over C(T). If A is an inductive limit of k direct sums of matrix algebras over C(T), then aspr(A) ≤ k. Indeed, for such a k direct sum, spr(⊕ k j=1Mnj (C(T))) = spr(⊕ k j=1(Mnj (C) ⊗ C(T))) = k∑ j=1 spr(Mnj (C) ⊗ C(T)) ≤ k∑ j=1 (spr(Mnj (C)) + spr(C(T))) = k. 82 Takahiro Sudo CUBO 10, 2 (2008) In particular, if T 2 θ is a simple noncommutative 2-torus, then it is an inductvie limit of 2 direct sums of matrix algebras over C(T). Hence, aspr(T2 θ ) ≤ 2. Proposition 2.4. Let X be a locally compact Hausdorff space with dim X finite. Then aspr(C0(X)) ≤ dim X. Proof. Note that X can be viewed as a projective limit of the product spaces [0, 1]n, where dim X = n. Thus, C0(X) is an inductive limit of C([0, 1] n ). Since C([0, 1]n) ∼= ⊗nC([0, 1]), we obtain the conclusion. 2 Remark. There exists a locally compact Hausdorff space X with dim X = 1 but dim X+ = 0, where X+ is the one-point compactification of X. Thus, aspr(C0(X)) = 1, but spr(C0(X)) = spr(C(X + )) = 0, where C0(X) + ∼= C(X+). Also, aspr(C(X+)) = 0. Remark. More fundamental properties for the approximate spectral rank could be obtained as the spectral rank in Section 1, but their details would be considered somewhere else. Received: February 2007. Revised: April 2008. References [1] L.G. Brown and G.K. Pedersen, C∗-algebras of real rank zero, J. Funct. Anal. 99 (1991), 131–149. [2] J. Dixmier, C∗-algebras, North-Holland (1977). [3] G.K. Pedersen, C∗-Algebras and their Automorphism Groups, Academic Press (1979). [4] M.A. Rieffel, Dimension and stable rank in the K-theory of C∗-algebras, Proc. London Math. Soc. 46 (1983), 301–333. [5] T. Sudo, A topological rank for C∗-algebras, Far East J. Math. Sci. 15 (1) (2004), 71–86. [6] T. Sudo, A covering dimension for C∗-algebras, Cubo A Math. J. 8 (1) (2006), 87–96. [7] W. Winter, Covering dimension for nuclear C∗-algebras, J. Funct. Anal. 199 (2003), 535–556. N6