SINGLY GENERATED C*-ALGEBRAS

ABSORBING SOME C*-ALGEBRA

Ryo Ichihara^{†} and Masaru Nagisa^{‡}

† Department of Mathematics Nara National College of Technology

‡ Department of Mathematics and Informatics Faculty of Science, Chiba University

Abstract. Let A be a UHF C*-algebra or a purely infinite, simple, unital, nuclear, separable C*-algebra. Then we have that the C*-tensor product A ⊗ B of A and B is singly generated for arbitrary unital, separable C*- algebra B.

1. Introduction

Let A be a unital, separable C*-algebras. We call a C*-algebra A |I|-generated if there exists a family {x_{i}}_{i∈I} of elements in A such that polynomials of x_{i}, x_{i}^{∗} (i ∈ I) with complex coefficients and no constant term are dense in A and we denote A = C^{∗}(x_{i}; i ∈ I). If A is generated by a self-adjoint elements a and a unitary element u in A, then A is generated by u(a + 2∥a∥), that is, A is singly generated. Needless to say, if A is generated by two self-adjoint elements, then A is singly generated. This fact implies that the following C*-algebras are singly generated: C(Ω)⋊ℤ (Ω is homeomorphic to a compact subspace of ℝ), C(Ω)⋊G (Ω is homeomorphic to a compact subspace of 𝕋 and G is an abelian group generated by countable elements with finite order), the reduced group C*-algebra of ℤ_{n} ∗ ℤ_{m} (n, m ∈ ℕ) ([4]), the free product C[0, 1] ∗ C[0, 1] of C[0, 1]’s by the usual Lebesgue measures and etc.

Since C[0, 1] ∗ C[0, 1] is σ-weakly dense in W_{r}^{∗}(F_{2}) ([16]), W_{r}^{∗}(F_{2}) is singly generated as a W*-algebra. But we do not know whether C_{r}^{∗}(F_{2}) is singly generated or not. We remark that the abelian C*-algebra C(𝕋) ⊗ C(𝕋) which is generated by two unitaries is not singly generated.

The first author shows that (not necessarily irreducible) Cuntz-Krieger C*- algebras are singly generated ([7]). He also shows that rotation C*-algebras and some AF C*-algebras are singly generated ([7,8]).

In this paper we consider the case that a unital, separable C*-algebra which absorbs a UHF C*-algebra or a purely infinite, simple, nuclear C*-algebra. We shall show that, for any unital, separable C*-algebra B, the C*-tensor product A ⊗ B is singly generated, if A is a UHF C*-algebra or a purely infinite, simple, nuclear C*-algebra.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 46L05.

2 RYO ICHIHARA^{†} AND MASARU NAGISA^{‡}

2. Main Results

We denote by M_{n} the algebra of all the n × n complex matrices. A finite dimensional C*-algebra is completely noncommutative if it has no abelian central projections.

With the number of generators for a C*-algebra, M. A. Rieffel([13]) shows that M_{2n+1} ⊗A is singly generated if a unital C*-algebra A is n-generated. This result is extended in [10] and [8], and the next lemma is an application of the first author’s result([8]). For the convienience of readers, we give the outline of this proof.

Lemma 1. Let a C*-algebra A be unital and 2-generated and a finite dimensional C*-algebra F completely noncommutative. Then we have the C*-tensor product A ⊗ F is singly generated.

In particular, A ⊗ M_{2} is singly generated.

Proof. Let unital C*-algebras A_{1}, A_{2}, …, A_{n} be singly generated and x_{i} a generator of A_{i} for each i. We may assume that there exist complex numbers λ_{1}, λ_{2}, …, λ_{n} satisfying that

|λ_{i}| > 1, Sp(x_{i}) ⊂ U(λ_{i}, 1) and U(λ_{i}, 2) ∩ U(λ_{j}, 2) = ϕ (i ≠ j),

where U(λ, a) means {z ∈ ℂ| |λ−z| < a}. Then we can show that A_{1}⊕A_{2}⊕⋯⊕A_{n} is singly generated. In fact, if we define

x = x_{1} ⊕ x_{2} ⊕ ⋯ ⊕ x_{n} ∈ A_{1} ⊕ A_{2} ⊕ ⋯ ⊕ A_{n},

then Sp(x) is contained in the disjoint union of U(λ_{i}, 1)’s. So we can get

I ⊕ 0 ⊕ ⋯ ⊕ 0, 0 ⊕ I ⊕ ⋯ ⊕ 0, ⋯, 0 ⊕ ⋯ ⊕ 0 ⊕ I ∈ C^{∗}(x),

by the holomorphic function calculas for x. Therefore x is a generator of A_{1} ⊕ A_{2} ⊕ ⋯ ⊕ A_{n}.

To prove the single generation of A ⊗ F, it is enough to show that A ⊗ M_{n} is singly generated for a 2-generated C*-algebra A. Then we have that

z = h ⊗ e_{11} + k ⊗ e_{12} + y ⊗ e_{22} + ∑_{i=2}^{n−1} I ⊗ e_{i,i+1}

is a generator of A ⊗ M_{n}, where {e_{i,j}} is a system of matrix units for M_{n} and x and y are generators of A such that h = (x + x^{∗})/2, k = (x − x^{∗})/2i are positive invertible,

Sp(y) ⊂ U(λ, 1) and Sp(h) ∩ U(λ, 2) = ϕ

for some λ ∈ ℂ with |λ| > 1. By the holomorphic function calculas for z we have I ⊗ e_{11}, I ⊗ e_{22} ∈ C^{∗}(z). So we get k ⊗ e_{12}, k^{2} ⊗ e_{11} and I ⊗ e_{12} ∈ C^{∗}(z) by the invertibility of k. This implies

a ⊗ e_{11} and ∑_{i=1}^{n−1} I ⊗ e_{i,i−1} ∈ C^{∗}(z)

for any a ∈ A. So we have A ⊗ M_{n} = C^{∗}(z). □

Remark. In [18], it is mentioned that C^{∗}(F_{2}) ⊗ M_{4} is singly generated by using Voiculescu’s result ([17]). The above lemma says that C^{∗}(F_{2}) ⊗ M_{2} is singly generated and it is known that C^{∗}(F_{2}) is not singly generated since there exists a *-homomorphism from C^{∗}(F_{2}) onto C(𝕋) ⊗ C(𝕋).

Theorem 2. If A is a simple, infinitely dimensional, singly generated C*-algebra, then the maximal C*-tensor product A ⊗_{max} B is 2-generated for any unital, separable C*-algebra B.

In particular, so is the minimal (or spacial) C*-tensor product A ⊗_{min} B.

Proof. At first we consider the case A is unital. We can choose generators x in A and {y_{n}}_{n=1}^{∞} in B such that

A = C^{∗}(x), B = C^{∗}(y_{1}, y_{2}, ⋯) and ∥y_{n}∥ = 1.

Since A is inifinitely dimensional, we can choose a family {p_{n}}_{n=1}^{∞} of positive elements in A satisfying p_{i}p_{j} = 0 if i ≠ j. We set

s = x ⊗ I and t = ∑_{n=1}^{∞} \frac{1}{n}p_{n} ⊗ y_{n},

then s and t belong to A ⊗ B. Clearly A ⊗ I(= C^{∗}(s)) is contained in C^{∗}(s, t), so we have p_{n}^{2} ⊗ y_{n} = n(p_{n} ⊗ I)t ∈ C^{∗}(s, t). For any n, we can choose elements a_{i} and b_{i} (1 ≤ i ≤ m) in A such that

∑_{i=1}^{m} a_{i}p_{n}^{2}b_{i} = I,

by the simplicity of A. This means that I ⊗ y_{n} = ∑_{i}(a_{i} ⊗ I)(p_{n}^{2} ⊗ y_{n})(b_{i} ⊗ I) ∈ C^{∗}(s, t), so we have A ⊗_{max} B is generated by s and t.

In the case A is non-unital, we can show that the above operators s and t also generate the C*-algebra A ⊗_{max} B. In fact, for any positive element p ∈ A and any positive number ε, we can choose elements a_{i} and b_{i} (1 ≤ i ≤ m) in A such that

∥ ∑_{i=1}^{m} a_{i}p_{n}^{2}b_{i} − p∥ < ε.

This means that p ⊗ y_{n} ∈ C^{∗}(s, t) for any positive element p ∈ A and any n. So we have A ⊗_{max} B = C^{∗}(s, t). □

As an example of unital, simple, infinitely dimensional, singly generated C*- algebra, the followings are known: UHF C*-algebras([15]), Bunce-Deddens algebras, simple crossed product C*-algebras C(Ω)⋊_{α}ℤ ( Ω is a compact subspace of ℝ), the C*-algebra 𝕂 of all compact operators, the reduced free product C[0, 1]∗C[0, 1] by the usual Lebesgue measures, irrational rotation C*-algebras ([8]) and the Cuntz algebra O_{n} generated by n isometries. Single generation of 𝕂 and O_{n} follows from the fact

𝕂 ⊗ M_{n} ≅ 𝕂 and O_{n} ⊗ M_{n} ≅ O_{n} ([11])

for any n ≥ 2. We also notice that, for any n ≥ 2 and any UHF C*-algebra A, there exist k ≥ n and a UHF C*-algebra B such that

A ≅ M_{k} ⊗ B.

So we can get the following result:

Corollary 3. If A is a UHF C*-algebra , O_{n} or 𝕂, then the C*-tensor product A ⊗ B is singly generated for any unital, separable C*-algebra B.

As stated in the above, O_{n} has the property O_{n} ⊗ M_{n} ≅ O_{n}. But O_{∞} is not isomorphic to O_{∞} ⊗ M_{n} for any n(≥ 2) by the Kirchberg’s classification theorem([9]).

Theorem 4. The C*-tensor product O_{∞} ⊗ B is singly generated for any unital, separable C*-algebra B.

Proof. We can choose {y_{n}}_{n=1}^{∞} in B such that

\frac{1}{2^{n}} I < y_{n} < \frac{1}{2^{n−1}} I and B = C^{∗}(y_{1}, y_{2}, …),

where a < b means that b−a is positive and invertible. Then we define x ∈ O_{∞} ⊗B as follows:

x = ∑_{i=1}^{∞} \frac{1}{2^{i}} S_{i} ⊗ I(∑_{j=1}^{∞} P_{j} ⊗ y_{j} + I),

where P_{j} = S_{j}S_{j}^{∗}. Then we have

x^{∗}x = \frac{1}{3}(∑_{j=1}^{∞} P_{j} ⊗ y_{j} + I)^{2},

this implies that ∑_{j} P_{j} ⊗ y_{j} + I ∈ C^{∗}(x) and P_{j} ⊗ y_{j} ∈ C^{∗}(x). Since y_{j} is positive and invertible, we have P_{j} ⊗1 in C^{∗}(x). The element ∑ P_{j} ⊗y_{j} +I is also invertible, so we have

∑_{i=1}^{∞} \frac{1}{2^{i}} S_{i} ∈ C^{∗}(x).

Therefore S_{j} ⊗ I in C^{∗}(x) and I ⊗ y_{j} in C^{∗}(x), that is, C^{∗}(x) = O_{∞} ⊗ B. □

E. Kirchberg shows that A ⊗ O_{∞} is isomorphic to A for any purely infinite, simple, unital, nuclear, separable C*-algebra. So we can get the following result as an easy consequence of this result and Theorem 2:

Corollary 5. If A is a purely infinite, simple, unital, nuclear, separable C*-algebra, then the C*-tensor product A ⊗ B is singly generated for any unital, separable C*- algebra B.

Especially, A is singly generated.

The notion of real rank is introduced by L. G. Brown and G. K. Pedersen([2]). Recently N. C. Phillips([12]) shows that the real rank of A⊗B is less than 1 for any purely infinite, simple, unital, nuclear, separable C*-algebra A and any C*-algebra B. The authors have an interest for the relation of the real rank of a C*-algebra and the least number of its generators, that is, RR(A) ≤ 1 implies that A is singly generated or not. For n ≥ 2, we have the following examples:

RR(C(𝕋^{n})) = n, RR(C^{∗}(F_{2})) = ∞
RR(C([0, 1]^{n(8n+3)}) ⊗ M_{4n+2}) = n, RR(C^{∗}(F_{2}) ⊗ M_{2}) = ∞,

where C(𝕋^{n}) and C^{∗}(F_{2}) are not singly generated and C([0, 1]^{n(8n+3)}) ⊗ M_{4n+2} and C^{∗}(F_{2}) ⊗ M_{2} are singly generated ([1],[10]).

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† 1119 Shin-machi, Yamato-Koriyama, Nara 639-11 Japan

   E-mail address: ichihara@nara-k.ac.jp

‡ 1-33, Yayoi-Cho, Inage-Ku, Chiba 263 Japan

   E-mail address: nagisa@math.s.chiba-u.ac.jp