Goldie dimension for C*-algebras

Abstract

In this article, we introduce and study the notion of Goldie dimension for C*-algebras. We prove that a C*-algebra A has Goldie dimension n if and only if the dimension of the center of its local multiplier algebra is n. In this case, A has finite-dimensional center and its primitive spectrum is extremally disconnected. If moreover, A is extending, we show that it decomposes into a direct sum of n prime C*-algebras. In particular, every stably finite, exact C*-algebra with Goldie dimension, that has the projection property and a strictly full element, admits a full projection and a non-zero densely defined lower semi-continuous trace. Finally we show that certain C*-algebras with Goldie dimension (not necessarily simple, separable or nuclear) are classifiable by the Elliott invariant.

1. Introduction

Noetherian and/or Artinian C*-algebras as well as C*-algebras with Krull dimension are defined and studied in [20, 34, 35, 40, 41]. In this article, we define and study C*-algebras with Goldie dimension as a generalization of all of these classes (see Figure 1), and then extend the main results obtained in [41] and present some new results and applications.

Figure 1. Classes of C*-algebras with (and without) Goldie dimension.

In abstract algebra, Alfred Goldie used the notion of uniform modules to construct a measure of dimension for modules, now known as the Goldie dimension (or the uniform dimension) of the module. The basic idea is to measure the number of ‘building blocks’ in a direct sum of non-zero submodules of M. The Goldie dimension (unlike the Krull dimension) in some cases behaves like dimension of a vector space (e.g., $\Bbb C^n$ is a C*-algebra with Goldie dimension n, while it has Krull dimension 0). We refer the reader to [8, Chapters 1 and 2], [16], [17, Chapters 1 and 3], [18, Chapter 5], [25, Section 6] and [27, Chapter 2], for results on the Goldie dimension and extending (or, CS) modules, in algebraic settings.

Our main motivation for defining the Goldie dimension for C*-algebras is to bridge between algebraic and topological properties of a large class of non-simple C*-algebras. Here, we work with (closed) two-sided ideals instead of one-sided ideals (as in the algebraic setting). We show that C*-algebras with Goldie dimension share some basic properties of C*-algebras with Krull dimension (obtained in [41]) and their local multiplier algebra and primitive ideal space can be described. We show that the Goldie dimension of a C*-algebra (if exists) is the same as the dimension of the center of its local multiplier algebra. In particular, a C*-algebra with Goldie dimension has finite-dimensional center and is boundedly centrally closed (compare with [41, Corollary 2.9 and Proposition 2.11]). We also give a classification theorem (based on the Kirchberg–Phillips classification theorem) for a subclass of C*-algebras with Goldie dimension (see Corollary 2.11 and Theorems 2.9 and 2.20).

The paper is organized as follows. In $\S$ 2, we define a notion of the Goldie dimension for C*-algebras (Definition 2.2), which is shown to be well-defined (Lemma 2.1), and conclude that every C*-algebra with Goldie dimension has a closed essential ideal which is a finite direct sum of prime C*-algebras (extending [41, Theorem 2.8]). We study the perseverance of the Goldie dimension under passing to hereditary C*-subalgebras, extensions and Morita equivalence. We show that the Goldie dimension is not preserved under taking quotients (unlike the Krull dimension) and crossed products in general. We provide conditions for these to happen (Theorem 2.7(iv) and Corollary 2.13). We give necessary and sufficient conditions for a C*-algebra to be have Goldie dimension (Theorems 2.7(1), 2.9 and 2.24). One of the advantages of C*-algebras with Goldie dimension is that they have extremally disconnected primitive spectrum (Theorem 2.9) and they can have compact ideals, full projections and non-zero (densely defined lower semi-continuous) traces, under certain conditions (Theorem 2.14). We introduce complete-Goldie C*-algebras and “extending” C*-algebras, and show that all complete-Goldie graph C*-algebras have real rank zero, and all extending C*-algebras with Goldie dimension are decomposable into a finite direct sum of prime C*-algebras (Corollary 2.17, and Theorem 2.24(iii)). This shows that every Hilbert C*-module, over an AW*-algebra or an extending C*-algebra with Goldie dimension n, decomposes into a direct sum of n ideal submodules (as in the case of Hilbert spaces; see Theorem 2.26).

2. Goldie dimension for C*-algebras

In this section, we define the Goldie dimension for C*-algebras and obtain the main properties of C*-algebras with Goldie dimension.

In the following results, by an ideal, we always mean a closed two-sided ideal, unless otherwise specified. For an ideal I (resp. C*-subalgebra B) in a C*-algebra A, we write $I\unlhd A$ (resp. $B\leq A$). If I is an essential ideal in a C*-algebra A, then we write $ I\unlhd_{e} A$. We denote the center of a C*-algebra A by Z(A). Also, for a C*-algebra A, let Id(A) denote the lattice of closed ideals of A.

Let us first recall the notion of chain conditions for C*-algebras. Chain conditions on a C*-algebra are defined in its set of closed (two-sided) ideals. A C*-algebra A is called $\it{Noetherian}$ if it satisfies the ascending chain condition for closed ideals, that is, for any ascending chain $ I_1 \subseteq I_2 \subseteq I_3 \subseteq, \ldots $ of closed ideals of A, there is a positive integer n such that $ I_i = I_n $, for all $ i \geq n $. The dual notion to a Noetherian C*-algebra is that of an $\it{Artinian}$ C*-algebra, which satisfies the descending chain condition for closed ideals. Clearly, every C*-algebra with finitely many closed ideals is Noetherian and Artinian, including simple and finite-dimensional C*-algebras. On the other hand, there are infinitely many mutually non-isomorphic Noetherian (Artinian) C*-algebras with infinitely many closed ideals [20, 40]. Recall that a Noetherian (Artinian) topological space is a space that satisfies the ascending (descending) chain condition for its open subsets. In the Noetherian case, a topological space is Noetherian if every open set is compact. Let Prim(A) be the set of primitive ideals in a C*-algebra A. Then, Prim(A) is a topological space with the Jacobson (or hull-kernel) topology [9, 20]. In [20], it was shown that a C*-algebra is Noetherian (Artinian) if and only if Prim(A) is so as a topological space. Note that the definition of Notherian (and Artinian) C*-algebras differs from the definition of Noetherian (and Artinian) rings, which is defined on the set of one-sided ideals. Also the definition of a Noetherian C*-algebra differs from the definition of a Noetherian Banach algebra, which is defined by a chain condition on the set of closed left ideals. In the latter definition, every Noetherian Banach algebra is finite-dimensional. This is also true (under the new definition based on ideals, rather than left ideals) for commutative C*-algebras, but not in general (see [20, Theorem 3.1]). Note that being Artinian (resp. Noetherian) is stable under taking hereditary C*-subalgebras and it passes to closed ideals and quotients, and also it is preserved under extension and Morita equivalence (see [34, Lemma 2.2 and Corollary 2.5] and [35, Proposition 1.1]).

Recall that, the Krull dimension of a C*-algebra measures how close the C*-algebra is to being Artinian (see [41, Definition 2.1], for a detailed definition). The proof of the following lemma is similar to that of Theorem 2.8 in [41].

Lemma 2.1. Let A be a C*-algebra and it contains an essential ideal I of the form $\bigoplus_{i=1}^{n} I_i$, where each I_i is a prime C*-algebra. Then,

(i) Every direct sum of non-zero ideals of A has at most n summands.

(ii) Let $J=\bigoplus_{i=1}^{m} J_i$ be an ideal in A, where each J_i is a prime C*-algebra. Then, $J\unlhd_e A $ if and only if m = n.

Definition 2.2. A C*-algebra A is said to have $\it{Goldie\; dimension}$ n ($n\in \Bbb N$), writing $\mathrm{G.dim}(A)=n$, if it contains an essential ideal which is the direct sum of n prime C*-algebras. If such an ideal exists, we call it a G-${\it ideal}$ in A and say that A is a C*-algebra with Goldie dimension (or A has Goldie dimension, or A is a Goldie C*-algebra). Otherwise, we say A is a C*-algebra without Goldie dimension (or A does not have Goldie dimension).

Note that $\mathrm{G.dim}(A)$ is well-defined, by Lemma 2.1. See Examples 2.21, 2.23 and 2.25, for examples of G-ideals in certain C*-algebras with Goldie dimension. We use the convention that for a C*-algebra A, $\mathrm{G.dim}(A)=0$ if and only if $A=\{0\}$. Also, every C*-algebra A with Krull dimension contains an essential ideal that is a finite direct sum of critical ideals [41, Theorem 2.8], here called a K-${\it ideal}$ in A.

Recall that the $\it{local\; multiplier\; algebra}$ $M_{loc}(A)$ of a C*-algebra A is the C*-direct limit of the multiplier algebras M(I), where I ranges over all (closed) essential ideals of A [2, Definition 2.3.1].

Remark 2.3. (a) A C*-algebra is prime if and only if $\mathrm{G.dim}(A)=1$. Indeed, if $\mathrm{G.dim}(A)=1$, then A contains an essential ideal I_A that is a prime C*-algebra. But in [2, Propositions 3.3.2 and 2.3.6(i)], it was shown that a C*-algebra is prime if and only if its local multiplier algebra is prime, and also the local multiplier algebra of a C*-algebra is isomorphic to the local multiplier algebra of any of its essential ideals. Thus $M_{loc}(A)\cong M_{loc}(I_A)$, and hence A is prime. The converse is clear.

(b) For an ideal I in a C*-algebra A with Goldie dimension n, $I\unlhd_{e} A$ if and only if $\mathrm{G.dim}(A)=\mathrm{G.dim}(I)$. Because, if $I\unlhd_{e} A$ and $\bigoplus_{i=1}^{n} I_i$ is a G-ideal in A, ($n \in \Bbb N$), then for each i, ($1\leq i \leq n$), $I\cap I_i \neq \{0\}$. Thus, $\bigoplus_{i=1}^{n} (I\cap I_i)$ is an essential ideal of prime C*-algebras in I, and so $\mathrm{G.dim}(I)$=n. For the converse, let $\mathrm{G.dim}(A)=\mathrm{G.dim}(I)$ but I is not essential. Then, there is an ideal J in A such that $I\cap J=\{0\}$. Also I contains an essential ideal which is the direct sum of n prime C*-algebras, say K. Then, $K\oplus J$ is a direct sum of n + 1 summands, which is a contradiction.

(c) Let A and B be C*-algebras with Goldie dimension. Then,

\begin{equation*}\mathrm{G.dim}(A\oplus B)= \mathrm{G.dim}(A) + \mathrm{G.dim}(B).\end{equation*}

Indeed, if $I_A=\bigoplus_{i=1}^{m} I^A_i$ and $I_B=\bigoplus_{i=1}^{n} I^B_i$ are G-ideals in A and B, respectively, then $I_A \oplus I_B \unlhd_e A\oplus B$, by [41, Lemma 2.6(iii)]. Thus, $\mathrm{G.dim}(A\oplus B)= m+ n$.

Definition 2.4. An ideal I in a C*-algebra A is said to be $\it{essentially\; closed}$, writing $I \unlhd_{e.c} A$, if whenever J is an ideal in A such that $I \unlhd_e J$ then I = J.

Every direct summand of a C*-algebra is essentially closed. In particular, for every C*-algebra A, the trivial ideals $\{0\}$ and A are essentially closed. Also, if an ideal in a C*-algebra is injective, then it is also essentially closed. Indeed, according to [19, Proposition 4.8], a C*-algebra is injective if and only if it has no proper essential extension, (a proper essential extension of an ideal I in a C*-algebra A means the existence of a C*-subalgebra B of A that $I \unlhd_e B$ but I ≠ B. Thus unlike a proper ideal in A, a proper essential extension can be equal to the C*-algebra A). It is worth mentioning that an essentially closed ideal is not necessarily an essential ideal (see Example 2.23(b)). In a C*-algebra, the only ideal that is both essentially closed and essential is the C*-algebra itself.

Lemma 2.5. Let A be a C*-algebra and I, J and K be ideals in A. Then, the following assertions hold: (i) (Modular Law) $I+(J\cap K) = (I+J) \cap (I+K)$. In particular, if $I\subseteq J$, then $I+(J\cap K) = J\cap (I+K)$.

(ii) If $I \unlhd J$ and $J/I \unlhd_e A/I$, then $J \unlhd_e A$.

(iii) If $I \unlhd J \unlhd_e A$ and $I \unlhd_{e.c} A$, then $J/I \unlhd_e A/I$.

Proof. For (i) and (ii), see [9, II.5.1.4(iv)] and [18, Proposition 5.6(c)]. For (iii), let $\bar{I} \unlhd A/I$. Then, there is an ideal $K \supseteq I$ in A such that $\bar{I}=K/I$. If $(J/I) \cap (K/I) =0$, then $(J\cap K)/I=0$, and so $J\cap K=I$. Since $J \unlhd_e A$, $I= (J\cap K)\unlhd_e (A\cap K)= K$. But $I \unlhd_{e.c} A$. Thus I = K, and hence $K/I=0$.

Lemma 2.6. The relation ‘$\unlhd_{e.c} $’ is transitive.

Proof. Let A be a C*-algebra and I, J and K be ideals in A such that $I \unlhd_{e.c} J$ and $J \unlhd_{e.c} K$. First, for $J\unlhd K$, there is an ideal J ^ʹ in K such that $(J \oplus J')\unlhd_e K$, by [41, Lemma 2.7]. Since $J \unlhd (J \oplus J')\unlhd_e K$ and $J \unlhd_{e.c} K$, Lemma 2.5(iii) shows that $((J \oplus J')/J) \unlhd_e (K/J)$. Thus

\begin{equation*}(J/I) \oplus ((I \oplus J')/I) = ((J \oplus J')/I) \unlhd_e (K/I),\end{equation*}

by Lemma 2.5(ii). Similarly, for $I\unlhd J$, there is an ideal I ^ʹ in J such that $(I \oplus I')\unlhd_e J$, and so $((I \oplus I')/I) \unlhd_e (J/I)$. Therefore,

\begin{align*} ((I\oplus I' \oplus J')/I) & = ((I\oplus I')/I) \oplus ((I\oplus J')/I)\\ & \unlhd_e (J/I) \oplus ((I\oplus J')/I)\\ & \unlhd_e (K/I). \end{align*}

Now, to show that $I \unlhd_{e.c} K$, let $I \unlhd_e L \unlhd K$. Since $I \cap (I'\oplus J')=0$, $L \cap (I'\oplus J')=0$. By Lemma 2.5(i), we have

\begin{equation*}L \cap (I\oplus I'\oplus J')= I \oplus (L \cap (I'\oplus J'))=I.\end{equation*}

This means that $(L/I) \cap ((I\oplus I'\oplus J')/I) =0$. Thus $L/I=0$, and so I = L.

For the following theorem, we use the concepts in Section 2 of [40]. For more details on the theory of graph C*-algebras, we refer the reader to [3].

Theorem 2.7. (i) A C*-algebra has Goldie dimension if and only if it contains no infinite direct sum of non-zero ideals.

(ii) Every C*-algebra with Krull dimension has Goldie dimension. The converse does not necessarily hold.

(iii) Goldie dimension passes to hereditary C*-subalgebras and is preserved under extension and Morita equivalence of C*-algebras.

(iv) Let A be a C*-algebra and $I\lhd A$. Then, $I \unlhd_{e.c} A$ if and only if both I and $A/I$ have Goldie dimension and $\mathrm{G.dim}(A)=\mathrm{G.dim}(I) + \mathrm{G.dim}(A/I)$.

Proof. (i) If A has Goldie dimension n, then by Lemma 2.1(i), every direct sum of non-zero ideals of A has at most n summands. For the converse, we first show that A contains an ideal that is a prime C*-algebra. If not, since A is not prime, there are ideals I ₁ and $I_1'$ in A such that $I_1 \cap I_1'=\{0\}$. Also I ₁ is not a prime C*-algebra. Thus, there are ideals I ₂ and $I_2'$ in I ₁ such that $I_2 \cap I_2'=\{0\}$. By continuing this process, we will have a direct sum as $I_1 \oplus I_2 \oplus I_3 \oplus \ldots $, that is a contradiction. Therefore, A (and also each of its ideals) has an ideal that is a prime C*-algebra. Now, similar to the first paragraph of the proof of Theorem 2.8 in [41], we can show that A contains an essential ideal which is a finite direct sum of prime C*-algebras, and so has Goldie dimension.

(ii) Let A be a C*-algebra with Krull dimension. Then, A contains an essential ideal that is a finite direct sum of critical ideals. But every critical C*-algebra is prime [41, Lemma 2.3(vi)]. Thus, A has Goldie dimension. For the next assertion, let $S=\{n_i\}_{i\in \Bbb N}$ be a sequence of natural numbers $n_i \geq 2$, and consider the following directed graph $\mathcal{G}$:

Here, $(n_i)$ means that there are n_i loops at each vertex w_i. Since the graph $\mathcal{G}$ is downward directed and satisfies Condition (K) (and hence (L)), $C^*(\mathcal{G})$ is prime [1, Proposition 3.1]. Thus $\mathrm{G.dim}(C^*(\mathcal{G}))=1$. On the other hand, $C^*(\mathcal{G})$ is without Krull dimension. Indeed, having Krull dimension passes to quotients [41, Lemma 2.3(i)], while if we consider the saturated hereditary subset $H=\{w_1\}$, then [3, Theorem 2.1.6(b)] implies that,

\begin{equation*} C^*(\mathcal{G})/I_H \cong C^*(\mathcal{G}\setminus H) \cong \bigoplus_{{n_i} \in {S\setminus \{n_1\}}} \mathcal{O}_{n_i},\end{equation*}

where $\mathcal{O}_{n_i}$ is the Cuntz algebra, and so $ C^*(\mathcal{G})/I_H $ does not have Krull dimension.

(iii) We first show that if B is a hereditary C*-subalgebra of a C*-algebra A with Goldie dimension, then $\mathrm{G.dim}(B)\leq \mathrm{G.dim}(A)$.

Suppose that $L:=\overline{ABA}$ (the closed two sided ideal generated by B) and $Id_L(A)$ is the set of closed ideals in A contained in L. Then, there is a bijection as follows:

\begin{equation*} \Phi: Id(B) \to Id_L(A); \; J \mapsto \overline{AJA},\end{equation*}

(see [9, p. 90]), hence $Id(B)\cong Id_L(A)\cong Id(L)$. But $\mathrm{G.dim}(L)\leq \mathrm{G.dim}(A)$, by (i). Now, since L has Goldie dimension, so is B.

For the second assertion, let I be an ideal in a C*-algebra A. We show that if I and $A/I$ have Goldie dimension, then $\mathrm{G.dim}(A) \leq \mathrm{G.dim}(I) + \mathrm{G.dim}(A/I)$.

According to [41, Lemma 2.7], there is an ideal K in A such that $I\oplus K \unlhd_e A$, and

\begin{equation*}K\cong K/(I\cap K) \cong (I\oplus K)/I \leq A/I,\end{equation*}

(see [9, p. 84]). Thus,

\begin{align*} \mathrm{G.dim}(A) & = \mathrm{G.dim}(I\oplus K)\\ & = \mathrm{G.dim}(I) +\mathrm{G.dim}(K)\\ & \leq \mathrm{G.dim}(I) +\mathrm{G.dim}(A/I). \end{align*}

The last assertion holds, because two Morita equivalent C*-algebras have isomorphic lattices of closed ideals [36, Theorem 3.22], and so have the same Goldie dimension.

(iv) Note first that there is an ideal Z in A such that $I\oplus Z \unlhd_e A$. Furthermore, I and Z have Goldie dimension, by (iii).

Now, let $I \unlhd_{e.c} A$. Then since $I\unlhd I\oplus Z \unlhd_e A$, by Lemma 2.5(iii), $((I\oplus Z)/I) \unlhd_e (A/I)$. Thus,

\begin{equation*}\mathrm{G.dim}(A/I) = \mathrm{G.dim}((I\oplus Z)/I)=\mathrm{G.dim}(Z/(I\cap Z))=\mathrm{G.dim}(Z),\end{equation*}

hence,

\begin{equation*}\mathrm{G.dim}(A) =\mathrm{G.dim}(I\oplus Z) = \mathrm{G.dim}(I) + \mathrm{G.dim}(A/I).\end{equation*}

For the converse, suppose that $I\unlhd_e L \unlhd A$. We show that I = L. Since $I\unlhd A$ and $((I\oplus Z)/I) \unlhd (A/I)$, both I and $(I\oplus Z)/I$ have Goldie dimension and $\mathrm{G.dim}(I\oplus Z)\leq \mathrm{G.dim}(I) + \mathrm{G.dim}((I\oplus Z)/I)$, by the proof of (ii). Thus,

\begin{align*} \mathrm{G.dim}(I) + \mathrm{G.dim}(A/I) & = \mathrm{G.dim}(A) \\ & =\mathrm{G.dim}(I\oplus Z) \\ & \leq \mathrm{G.dim}(I)+\mathrm{G.dim}((I\oplus Z)/I). \end{align*}

Therefore, $\mathrm{G.dim}((I\oplus Z)/I) = \mathrm{G.dim}(A/I)$, and hence $((I\oplus Z)/I)\unlhd_e (A/I)$, by Remark 2.3(b). Since $L\cap Z =\{0\}$,

\begin{equation*}((I\oplus Z)/I) \cap (L/I) = ((I\oplus Z)\cap L)/I = (I\oplus (Z\cap L))/I =\{0\},\end{equation*}

by Lemma 2.5(i). This shows that $(L/I)=\{0\}$, and so I = L.

In particular, all the C*-algebras introduced in [41, Example 3.2(a–j)] have Goldie dimension.

Remark 2.8. (a) The graph C*-algebra $C^*(\mathcal{G})$, introduced in Theorem 2.7(ii), shows that Goldie dimension is not preserved under quotients.

(b) Let A be a C*-algebra with a non-trivial essentially closed ideal I. Then, $ \mathrm{G.dim}(A)=2$ if and only if both I and $A/I$ are prime C*-algebras. In fact, if $ \mathrm{G.dim}(A)=2$, then since $I\unlhd_{e.c} A$, Theorem 2.7(iv) shows that $\mathrm{G.dim}(I) + \mathrm{G.dim}(A/I)=2$. Since I is non-trivial, we conclude that $\mathrm{G.dim}(I)=1$ and $\mathrm{G.dim}(A/I)=1$. The converse is clear.

(c) Let A be a C*-algebra A with Goldie dimension, B be a simple C*-algebra, with A or B nuclear. Then, $A\otimes B$ has Goldie dimension (because $Id(A)\cong Id(A\otimes B)$). In particular, $A\otimes \Bbb K$ and $M_n(A)$ have Goldie dimension.

Let A be a C*-algebra, $\mathcal{I}_{ce}$ be the set of all closed essential ideals in A and,

\begin{equation*}C(A)={\mathrm{alg}\negthinspace-\negthinspace\underrightarrow{\mathrm{lim}}}_{I\in \mathcal{I}_{ce}} C(Prim(I))\;\, \mbox{and}\;\, C_b(A)={\mathrm{alg}\negthinspace-\negthinspace\underrightarrow{\mathrm{lim}}}_{I\in \mathcal{I}_{ce}} C_b(Prim(I)).\end{equation*}

Then, C(A) (resp. $C_b(A)$) is called the (resp. $\it{bounded}$) $\it{extended\; centroid}$ of A and we have $C(A)= Z(Q_s(A))$ (resp. $C_b(A)= Z(Q_b(A))$), where $Q_s(A)$ (resp. $Q_b(A)$) is the (resp. bounded) symmetric algebra of quotients of A (see [2, Proposition 2.2.5 and Theorem 2.2.8]).

Theorem 2.9. Let A be a C*-algebra and $n\in \Bbb N$. Then, $\mathrm{G.dim}(A)=n$ if and only if $Z(M_{loc}(A))\cong \Bbb C^n$.

Proof. Let $\mathrm{G.dim}(A)=n$ and $I_A=\bigoplus_{i=1}^{n} I_i$ be a G-ideal in A. Then,

\begin{equation*} Z(M_{loc}(A))\cong Z(M_{loc}(I_A)) \cong \bigoplus_{i=1}^{n} Z(M_{loc}(I_i))\cong \Bbb C^n,\end{equation*}

(see [2, Propositions 2.3.6 and 3.3.2]).

For the converse, let $Z(M_{loc}(A))\cong \Bbb C^n$ and C(A) be the extended centroid of A. According to [8, Theorem 10.3.41], $Z(M_{loc}(A))\cong \Bbb C^n$ if and only if $C(A) \cong \Bbb C^n$, if and only if A has exactly n minimal prime ideals, say $P_1, P_2, \ldots, P_n$. Since C(A) is finite dimensional,

\begin{equation*}C(A)=C_b(A)=Z(M(A))=Z(M_{loc}(A)),\end{equation*}

see [2, Propositions 2.2.13 and pp. 71 and 94]. Thus, A is boundedly centrally closed. But a C*-algebra A is boundedly centrally closed if and only if for every ideal J in A, $\mathrm{ann}_A(J)=eA$, for a projection $e \in Z(M(A))$ [2, Remark 3.2.7]. Now, let $e_1, e_2, \ldots, e_n$ be the minimal central projections in $M_{loc}(A)$. Then, we can write (by changing the index, if necessary),

\begin{equation*}\mathrm{ann}_{A}(P_1)=e_1A, \; \mathrm{ann}_{A}(P_2)=e_2A, \ldots,\; \mathrm{ann}_{A}(P_n)=e_nA.\end{equation*}

Let $J_i:=\mathrm{ann}_{A}(P_i)=e_iA$, $ i\in \{1, 2, \ldots, n\}$. In this case, $J_1, J_2 , \ldots, J_n$ are non-zero ideals in A. Since

\begin{equation*}M_{loc}(J_i)=e_iM_{loc}(A),\end{equation*}

and every projection e_i is minimal, we have $Z(M_{loc}(J_i))=\Bbb C$, and so every J_i is a prime C*-algebra [2, Proposition 3.3.2]. Since $\bigoplus_{i=1}^{n} J_i \unlhd_e A$, we conclude that A has Goldie dimension n.

A C*-algebra A is $\it{boundedly\; centrally\; closed}$ if $Z(M(A))=Z(M_{loc}(A))$; see [2, Definition 3.2.1 and Proposition 3.2.3]. Also, a topological space is $\it{extremally\; disconnected}$ if the closure of any open subset is still an open subset, equivalently, every bounded continuous complex-valued function on a dense open subset can be (uniquely) extended to a bounded continuous function on the whole space (see, [15, Section 1H] and the proof of Proposition 3.2.4 [2]). Note that, in [2, Proposition 3.2.4], it was shown that a C*-algebra A is boundedly centrally closed if and only if Prim(A) is extremally disconnected. Moreover, in the proof of above theorem, it was observed that for a C*-algebra A with Goldie dimension, we have,

\begin{equation*}Z(M(A))=Z(M_{loc}(A)).\end{equation*}

Therefore, the following corollary holds.

Corollary 2.10. Let A be a C*-algebra with Goldie dimension. Then, Prim(A) is extremally disconnected.

This result is an extension of [41, Proposition 2.11(i)] (without assumption of having a full projection).

Let us denote denote the class of all C*-algebras for which $M_{loc}(A)=A$ by $\mathcal{M}_l$. All simple and unital C*-algebras, AW*-algebras, and any finite direct sum of these, belong to $\mathcal{M}_l$.

Corollary 2.11. Every C*-algebra with Goldie dimension has finite dimensional center. The converse also holds when the C*-algebra belongs to $\mathcal{M}_l$.

Proof. Let A be a C*-algebra with Goldie dimension n. Then, $ Z(A) \subseteq Z(M_{loc}(A))\cong \Bbb C^n,$ by Theorem 2.9 and [2, Lemma 3.2.2(i)]. Thus, Z(A) is finite dimensional.

If $\mathrm{dim}(Z(A))=n$ and $A \in \mathcal{M}_l$, then $\mathrm{dim}(Z(M_{loc}(A)))=n$, and hence $\mathrm{G.dim}(A)=n$, by Theorem 2.9.

The converse direction in the above corollary does not hold in the general. For instance, if $A:= \bigoplus_{i=1}^\infty A_i$, where $A_i=\Bbb K$, and A ¹ is the unitization of A, then $Z(A)=\{0\}$ and $Z(A^1)=\Bbb C$, but A does not have Goldie dimension, by Theorem 2.7(i).

As an application, an AW*-algebra has Goldie dimension if and only if it has finite-dimensional center. Also, a unital C*-algebra A with Prim(A) Hausdorff, has such a property. Indeed, if A has also finite-dimensional center, then Prim(A) is finite, because $C(Prim(A)) \cong Z(A)$, by the Dauns-Hofmann theorem [36, Theorem A.34].

Note that for the C*-algebra $A:= \bigoplus_{i=1}^\infty A_i$, where for all i, $A_i=\Bbb K$, we have

\begin{equation*}M_{loc}(A)=M(A)= \bigoplus_{i=1}^\infty \Bbb B,\end{equation*}

and hence $Z(M_{loc}(A))$ is infinite-dimensional, which of course is confirmed by Theorem 2.9.

Recall that a C*-algebra A is of type I if the von Neumann algebra generated by the range of any (non-degenerate) representation is of type I (i.e., in its lattice of projections, every non-zero central projection dominates a non-zero abelian projection).

Corollary 2.12. (i) Let X be a compact and Hausdorff space. Then, C(X) has Goldie dimension if and only if $|X| \lt \infty$.

(ii) A C*-algebra of type I has Goldie dimension if and only if it has Krull dimension zero.

Proof. (i) Suppose C(X) has Goldie dimension. By Corollary 2.11, every commutative C*-algebra with Goldie dimension is finite-dimensional. Thus, the cardinal of X is finite. Conversely, if $|X| \lt \infty$, then $Prim(C(X))\cong X$ [9, p. 60], and hence C(X) is Artinian. Now since $K.dim(C(X))=0$, C(X) has Goldie dimension, by Theorem 2.7(ii).

(ii) Let A be a C*-algebra of type I and has Goldie dimension. A C*-algebra is of type I if and only if it is Morita equivalent to a commutative C*-algebra [7, Theorem 2.2]. Furthermore, Goldie dimension is preserved under Morita equivalence of C*-algebras. Thus, there is a finite dimensional C*-algebra B such that $A \sim_M B$, and so Prim(A) is finite. This implies that $K.dim(A)=0$. The converse is clear.

In particular, $C(\mathbb{T})$ is a C*-algebra without Goldie dimension.

Note that Goldie dimension is not preserved under taking crossed products in general. Let α denote the action of the group $\Bbb Z$ on the unit circle $\mathbb{T}$, by rotations through multiples of a fixed angle $2\pi\theta$, where θ is irrational, and $ ( C(\mathbb{T})\rtimes_{\alpha} \Bbb Z , \mathbb{T} , \widehat{\alpha} )$ be the dual system of $(C(\mathbb{T}) , \Bbb Z , \alpha)$ (see [46, p. 190]). The irrational rotation C*-algebra $A= C(\mathbb{T})\rtimes_{\alpha} \Bbb Z$ is a simple C*-algebra [9, Example II.10.4.12(i)]. On the other hand,

\begin{equation*}\mathrm{G.dim}(A \rtimes_{\widehat{\alpha}} \mathbb{T}) = \mathrm{G.dim}(C(\mathbb{T})\otimes K(L^2(\Bbb Z))) =\mathrm{G.dim}(C(\mathbb{T})),\end{equation*}

by the Takai duality theorem [46, Theorem 7.1], and so $A \rtimes_{\widehat{\alpha}} \mathbb{T}$ does not have Goldie dimension.

Corollary 2.13. Let G be a finite group acting by $\alpha: G \to Aut(A)$ on a unital C*-algebra A with Goldie dimension. If α has Rokhlin property, then the maximal crossed product $A \rtimes_{\alpha} G$ and the fixed point algebra A ^α have Goldie dimension.

Proof. Since A is unital, G is finite and α is a Rokhlin action, the surjective map

\begin{equation*} Id(A \rtimes_\alpha G) \to Id(A)^G : J \mapsto J \cap A,\end{equation*}

is a lattice isomorphism by [42, Section 1.2 and Theorem 1.30]. Thus, $A \rtimes_{\alpha} G$ has also Goldie dimension, by Theorem 2.7(i). Also, [9, Theorem II.10.4.18] implies that there is a projection $p \in M(A \rtimes_{\alpha} G)$ such that

\begin{equation*}A^\alpha\cong p(A \rtimes_{\alpha} G)p,\end{equation*}

and hence A ^α has Goldie dimension, by Theorem 2.7(iii).

For a subset S of a C*-algebra A, $\overline{ASA}$ denotes the closed two sided ideal generated by S, and we simply write $\overline{AaA}$, when $S=\{a\}$. An element $a\in A$ is called $\it{full}$ if $A=\overline{AaA}$ ([9], p. 91). Let A be a C*-algebra, $a \in A^+$ and ɛ > 0. Then $(a-\varepsilon)_{+}:= h_{\varepsilon}(a)$, where $h_{\varepsilon}: \Bbb R^+\to \Bbb R^+$ is given by:

\begin{equation*}h_{\varepsilon}(t)=\max\{(t-\varepsilon) , 0 \}.\end{equation*}

Given $a, b \in A^+$, we say that a is Cuntz subequivalent to b (and write $ a \precsim b$), if there is a sequence $\{x_k\}_{k=1}^\infty \subseteq A$ such that $x_k^* b x_k \to a$ in norm. We say that a and b are Cuntz equivalent (and write $a\sim_{cu} b$), if $ a \precsim b$ and $ b \precsim a$ [23]. An element $a\in A$ is $\it{strictly\; full}$ if $(a-\varepsilon)_{+}$ is full for some ɛ > 0, and hence for all sufficiently small ɛ > 0 [28, p. 46].

An ideal J in a C*-algebra A is called a $\it{compact\; ideal}$ if whenever $(J_\lambda)_{\lambda\in \Lambda}$ is an increasing net of ideals in A such that $A=\overline{\bigcup_{\lambda\in \Lambda} J_\lambda}$, then $J=J_\lambda$ for some λ (see [32, Remark 2.2]). A C*-algebra A has the $\it{projection\; property}$, if each of its ideals has an increasing approximate unit consisting of projections (see [31, Definition 1]). Note that for the projection property, according to [11, Definition 4.8], the assumption of the approximate unit to be increasing is necessary.

Denote by $T_{lsc}(A)$ the cone of linear traces on a C*-algebra A whose domain is the Pedersen ideal Ped(A) of A. We can identify $T_{lsc}(A)$ with the set of densely defined lower semi-continuous traces on A (see [39, Definition 2.12]). Let S ^∘ (resp. S^c) denote the interior (resp. the complement) of a subset S of Prim(A) (in the hull-kernel topology).

Theorem 2.14. Let A be a C*-algebra with Goldie dimension containing a strictly full element. Then, the following assertions hold:

(i) Let Prim(A) be non-Hausdorff and $J\unlhd A$. Then, there is a compact ideal $\hat{J}$ in A such that:

\begin{equation*}\bigcap_{I\in U} I=J \subseteq \hat{J} = \bigcap_{I\in U^\circ} I; \; \mbox{where} \;\, U:=\{I \in Prim(A) \, | \, J \subseteq I \}\; \mbox{and} \;\, U^\circ \neq \emptyset,\end{equation*}

and $\hat{J}$ has a strictly full element and a maximal ideal.

(ii) Let A be exact. If A is purely infinite (resp. stably finite and has the projection property), then it has a full projection p (resp. q) and $T_{lsc}(A)=\{0\}$ (resp. $T_{lsc}(A)\neq \{0\}$ and for any $\tau \in T_{lsc}(A)$, $\tau(q)\neq 0$).

Proof. First note that there is a lattice isomorphism ϕ between the ideal lattice Id(A) of A and the lattice $\Bbb{O}(Prim(A))$ of open subsets of Prim(A) such that corresponding to an ideal $J\in Id(A)$, we have the following open set,

\begin{equation*}\mathrm{hull}(J)^c =\{I \in Prim(A) \, | \, J \subseteq I \}^c = Prim(A/J)^c= Prim(J),\end{equation*}

and, conversely, corresponding to an open set $V\subseteq Prim(A)$, we have the following ideal,

\begin{equation*}\mathrm{ker}(V^c)=\bigcap_{I\in V^c} I,\end{equation*}

(see [32, Remark 2.2] and [13, Proposition 3.2.1]).

Moreover, Prim(A) is compact. Indeed, let x be a strictly full element in A, $(J_\lambda)_{\lambda\in \Lambda}$ be an increasing net of ideals in A such that $A=\overline{\bigcup_{\lambda\in \Lambda} J_\lambda}$. Then, $(x-\varepsilon)_{+}$ is a full positive element in A for some ɛ > 0, and since

\begin{equation*}(x-\varepsilon)_{+} \in Ped(A) \subseteq \bigcup_{\lambda \in \Lambda} J_\lambda,\end{equation*}

there is λ ₀ such that $(x-\varepsilon)_{+} \in J_{\lambda_0}$. Hence $J_{\lambda_0}=A$. This implies that A is compact as an ideal, and so Prim(A) is compact.

For (i), since Prim(A) is extremally disconnected, $\overline{Prim(J)}$ is a closed and open subset. But Prim(A) is compact. Thus, $\overline{Prim(J)}$ is compact-open, and hence its corresponding ideal, say $\hat{J}$, is compact and we have,

\begin{equation*}\hat{J}= \mathrm{ker}((\overline{Prim(J)})^c)=\bigcap_{I\in (\overline{Prim(J)})^c} I\end{equation*}

and

\begin{equation*}(\overline{Prim(J)})^c=((Prim(J))^c)^\circ=\{I \in Prim(A) \, | \, J \subseteq I \}^\circ=U^\circ.\end{equation*}

Of course, corresponding to the open set Prim(J) we have

\begin{equation*}J=\mathrm{ker}((Prim(J))^c)= \bigcap_{I\in U} I.\end{equation*}

Note that $U^\circ\neq \emptyset$. Otherwise $\overline{Prim(J)}^c =\emptyset$, and so $\overline{Prim(J)}=Prim(A).$ On the other hand, according to [15, Section 6M(2)], a compact space X is extremally disconnected if and only if $X= \beta Y$ for every dense subspace Y, where $\beta Y$ is the Stone-Čech compactification of Y. Thus,

\begin{equation*}Prim(A)= \beta Prim(J),\end{equation*}

and hence Prim(A) is Hausdorff, which is a contradiction. Now, since $U^\circ\neq \emptyset$, we have $J\subseteq \hat{J}$. Let $(z_\lambda)_{\lambda\in \Lambda}$ be an approximate unit for $\hat{J}$, and $\hat{J}_\lambda:=\overline{\hat{J}z_\lambda \hat{J}}$. Then, $\phi(\hat{J}_\lambda)=Prim(\hat{J}_\lambda)$ and

\begin{equation*}\hat{J}=\overline{\bigcup_{\lambda \in \Lambda} \hat{J}_\lambda}=\overline{\sum_{\lambda \in \Lambda} \hat{J}_\lambda}.\end{equation*}

Since ϕ is a lattice isomorphism, $\phi(\hat{J})= \bigcup_{\lambda \in \Lambda} Prim(\hat{J}_\lambda)$. Since $\phi(\hat{J})=Prim(\hat{J})$ is compact, there are $\lambda_1, \lambda_2, \ldots, \lambda_n$ in Λ such that, $\phi(\hat{J})= \bigcup_{i=1}^n Prim(\hat{J}_{\lambda_i})$, and,

\begin{equation*}\hat{J}= \overline{\sum_{i=1}^n \hat{J}_{\lambda_i}}= \sum_{i=1}^n \overline{\hat{J}z_{\lambda_i} \hat{J}}.\end{equation*}

Thus, $z=\sum_{i=1}^n z_{\lambda_i}$ is a full positive element in $\hat{J}$.

According to [33, Proposition 4.4.4] (or [9, Proposition II.6.5.5(i)]), for $z\in \hat{J}$, there is a lower semicontinuous real function $\check{z}$ on $Prim(\hat{J})$, given by:

\begin{equation*} I \mapsto \| z/ \mathrm{ker}(\pi)\|,\end{equation*}

where $I\unlhd \hat{J}$ and π is an irreducible representation of $\hat{J}$ such that $I=\mathrm{ker}(\pi)$, by [33, Proposition 4.4.2] (recall that unitarily equivalent representations have the same kernel).

Since $Prim(\hat{J})$ is compact, by [14, Theorem 7.4.14] (which states that any lower semicontinuous function on a compact space attains its minimum value), there is $K \in Prim(\hat{J})$ such that,

\begin{equation*}\check{z}(K) =\| z/ \mathrm{ker}(\pi)\|= \mathrm{inf}_{I\in Prim(\hat{J})}\check{z}(I) \gt 0.\end{equation*}

Note that, since z is full, it does not belong to any proper (primitive) ideal and therefore does not belong to the kernel of any non-zero irreducible representation, and hence $\| z/ \mathrm{ker}(\pi)\| \gt 0.$ Since

\begin{equation*}z=\mathrm{lim}_{n\rightarrow\infty} (z-1/n)_{+}.\end{equation*}

and $\check{z}_n\to \check{z}$ uniformly, where $z_n=(z-1/n)_{+}$, for $0 \lt \varepsilon \lt \check{z}(K)$, there is $n_0\in \Bbb N$ such that, for all $n \geq n_0$ and all $I\in Prim(A)$,

\begin{equation*}| \| z_n/ \mathrm{ker}(\pi)\| - \| z/ \mathrm{ker}(\pi)\| | \lt \varepsilon,\end{equation*}

hence,

\begin{equation*}0 \lt \| z/ \mathrm{ker}(\pi)\| - \varepsilon \lt \| z_n/ \mathrm{ker}(\pi)\|.\end{equation*}

This implies that z_n is full in $\hat{J}$, for all $n \geq n_0$, and hence z a strictly full element in $\hat{J}$.

For the last assertion of (i), let Ω be the set of all proper ideals in $\hat{J}$ and Ω₀ be a chain in Ω. Since the set of all ideals in a C*-algebra is a complete lattice,

\begin{equation*}I_0:=\bigvee_{I\in \Omega_0} I=\overline{\bigcup_{I\in \Omega_0} I}\end{equation*}

is an ideal in $\hat{J}$. Since $\hat{J}$ is a compact ideal, $I_0 \subsetneq \hat{J}$ (because if $I_0 =\hat{J}$, then for an $I\in \Omega_0$ we have $\hat{J}= I$, that is a contradiction). Hence I ₀ is a proper ideal in $\hat{J}$, and in fact an upper bound for Ω₀. Thus, by Zorn’s Lemma, Ω has a maximal ideal. This implies that $\hat{J}$ has a maximal ideal.

For (ii), first let A be exact and purely infinite. Since x is a strictly full element in A, for some $\varepsilon_0 \gt 0$, $x \precsim (x-\varepsilon_0)_{+}.$ Now by the last two paragraphs of the proof of Proposition 2.7($i\Rightarrow ii$) in [32] (which does not indeed need the assumption of separability), there is a projection p in A such that $x \sim_{cu} p$, and so p is full in A. But $Prim(A)=Prim(A\otimes \Bbb K)$ and $A\otimes \Bbb K$ is purely infinite and has also a full projection, of the form $(p\otimes e)$, where e is a rank-one projection in $\Bbb K$. Moreover, in [39, Theorem 2.15] it was shown that an exact C*-algebra A with Prim(A) compact, admits a non-zero densely defined lower semi-continuous trace if and only if the stabilization of A does not contain a full properly infinite projection. Thus, we have $T_{lsc}(A)=\{0\}$.

Now for the second case, suppose that A is exact and stably finite and has the projection property. Then, A has an increasing approximate unit consisting of projections, say $(q_\lambda)_{\lambda\in \Lambda}$. Let $A_\lambda:=\overline{A q_\lambda A}$. Then, $(A_\lambda)_{\lambda\in \Lambda}$ is an increasing net of ideals in A. Since $A=\overline{\bigcup_{\lambda \in \Lambda} A_\lambda}$, we have $A = A_{\lambda_0},$ for some $\lambda_0\in \Lambda$. Thus, $q_{\lambda_0}$ (resp. $q_{\lambda_0}\otimes e$) is a full projection in A (resp. $A\otimes \Bbb K$). Note that since A is stably finite, every projection in $A\otimes \Bbb K$ is finite. Hence $T_{lsc}(A)\neq \{0\}$, again by [39, Theorem 2.15].

Also, suppose that $\tau \in T_{lsc}(A)$. Then, $\tau(q_{\lambda_0})\neq 0$. Otherwise, $q_{\lambda_0} \in \mathrm{ker}(\tau)$. On the other hand,

\begin{equation*}q_{\lambda_0} \in Ped(A) = \mathrm{dom}(\tau),\end{equation*}

where $\mathrm{dom}(\tau)$ is the domain of τ (see [39, Definition 2.12]). Now, since $q_{\lambda_0}$ is full in A, $\mathrm{ker}(\tau)=\mathrm{dom}(\tau)$ and hence τ is zero, which is a contradiction.

Definition 2.15. A C*-algebra A is said $\it{complete}$-$\it{Goldie}$, if all its quotients have Goldie dimension.

All C*-algebras with Krull dimension are complete-Goldie. The graph C*-algebra $C^*(\mathcal{G})$, introduced in Theorem 2.7(ii), is a Goldie C*-algebra, but it is not complete-Goldie.

Question 2.16. Is there a complete-Goldie C*-algebra that does not have Krull dimension?

Corollary 2.17. Let E be a graph and $A=C^*(E)$ be a complete-Goldie graph C*-algebra. Then E satisfies Condition (K). In particular, $RR(A) = 0$.

Proof. The C*-algebra $C(\mathbb{T})$ does not have Goldie dimension, by Corollary 2.11 (or Corollary 2.12). Furthermore, for a complete-Goldie C*-algebra, Goldie dimension is preserved under Morita equivalence of C*-algebras and passes to ideals and quotients, by Theorem 2.7(i) and Definition 2.22. The first assertion now follow from [40, Lemma 3.3]. The second assertion holds, because in [21, Theorem 2.5], it was shown that if E is a directed graph, then $C^*(E)$ satisfies Condition (K) if and only if the real rank of $C^*(E)$ is zero.

In particular, every purely infinite complete-Goldie graph C*-algebra A is $\mathcal{O}_\infty$-stable and have nuclear dimension one. Indeed, since $RR(A)=0$, A has an approximate unit consisting of projections. Furthermore, [24, Theorem 9.1] implies that a nuclear and separable C*-algebra with an approximate unit consisting of projections is $\mathcal{O}_\infty$-stable if and only if it is strongly purely infinite, and a separable C*-algebra of real rank zero is strongly purely infinite if and only if it is purely infinite. Recall that every graph C*-algebra is separable and nuclear [3, p. 65]. Thus $A\cong A\otimes\mathcal{O}_\infty$. Moreover, in [10, Theorem A], it was shown that every $\mathcal{O}_\infty$-stable, nuclear and separable C*-algebra, has nuclear dimension one. Therefore, A has nuclear dimension one. We refer the reader for the notion of nuclear dimension of a C*-algebra to [47].

Lemma 2.18. The local multiplier algebra of an Artinian, purely infinite and separable C*-algebra is purely infinite.

Proof. Let A be an Artinian, purely infinite and separable C*-algebra. Then, for an $n \in \Bbb N$, there are the 0-critical (or simple) C*-algebras A_i, ($1\leq i \leq n$), such that $A= \bigoplus_{i=1}^n A_i$, by [41, Lemma 2.3(v) and Theorem 2.8]. Thus $M_{loc}(A)= \bigoplus_{i=1}^n M_{loc}(A_i)$. But, for every i, ($1\leq i \leq n$), $M_{loc}(A_i)=M(A_i)$. Hence $M_{loc}(A)= M(A)$. By Zhang’s theorem [49, Theorem 1.2(ii)], every purely infinite and simple C*-algebra has real rank zero. Furthermore, in [48, Theorem 1.3(b)], it was shown that the corona algebra of a simple and σ-unital C*-algebra of real rank zero is purely infinite. Thus, every $M(A_i)/A_i$ is purely infinite. Now being purely infinite is preserved under extensions. Therefore, every $M_{loc}(A_i)=M(A_i)$ is also purely infinite. This shows that $M_{loc}(A)$ is purely infinite.

Recall that a Kirchberg algebra is a purely infinite, nuclear, separable and simple C*-algebra [38]. Let $\Sigma(A)=\{[p] \,|\, p\in P(A) \}$ be the dimension range $\Sigma(A)$ of a C*-algebra A [37, p. 124].

Lemma 2.19. For every countable direct sum A of Kirchberg algebras, we have $K_0(A)=\Sigma(A)$.

Proof. Let $A= \bigoplus_{\lambda \in \Lambda} A_\lambda$, where Λ is a countable direct set and every A_λ is a Kirchberg algebra, and Ω be the collection of finite subsets of Λ, directed by inclusion. Then

\begin{equation*}A= {\underrightarrow{\mathrm{lim}}}_{\mathcal{F}} \bigoplus_{\lambda \in \mathcal{F}} A_\lambda,\end{equation*}

where $\mathcal{F}\in \Omega$ [9, Example II.8.2.2(ii)]. According to [34, Proposition 3.4(ii)], every Noetherian, purely infinite and separable C*-algebra has the ideal property (i.e., its ideals are generated by thier projections). Moreover, the ideal property and purely infinite are preserved under inductive limits (see [30, Proposition 2.3] and [23, Proposition 4.18]). Also, since Λ is countable, A is separable. Thus, A is a purely infinite and separable C*-algebra with the ideal property. Now, [32, Lemma 3.8] implies that $K_0(A)=\Sigma(A)$.

Every Artinian C*-algebra A has a K-ideal J_A that is a direct sum of m simple (or 0-critical) ideals, by [41, Lemma 2.3(v) and Theorem 2.8]. On the other hand, every prime C*-algebra has only one (non-zero) simple ideal (if any). Thus, every Artinian and prime C*-algebra A has a unique simple (essential) ideal $\mathcal{J}_A$. Now, if A is a C*-algebra with Goldie dimension n and with an Artinian G-ideal $I_A = \bigoplus_{i=1}^n I_i$, then every direct summand I_i of I_A is Artinian and prime. Thus every I_i has a unique simple ideal $\mathcal{J}_i$. We call the ideal $\mathcal{J}_{A}:= \bigoplus_{i=1}^n \mathcal{J}_i$ a G ₀-$\it{ideal}$ in A. Of course, $\mathcal{J}_{A}\unlhd_e A$, by [41, Lemma 2.6].

For each class of C*-algebras that can be classified, an appropriate Elliott invariant is used, commonly shown as $Ell(.)$. In the following theorem, using Lemma 2.19, we set,

\begin{equation*}Ell(.):= (K_0(.), K_1(.))=(\Sigma(.), K_1(.)), \end{equation*}

for the stable case and,

\begin{equation*}Ell(.):= (K_0(.), [1]_0, K_1(.))= (\Sigma(.), [1]_0, K_1(.)),\end{equation*}

for the unital case.

Theorem 2.20 Let A and B be C*-algebras with Goldie dimension n, each containing an Artinian G-ideal. Suppose that all direct summands of G ₀-ideals $\mathcal{J}_{A}$ and $\mathcal{J}_{B}$ are separable, nuclear, purely infinite and in the UCT class and for $n\geq 2$, are also stable and with non-zero simple K_j-groups, $j=0,1$. Then, $Ell(\mathcal{J}_{A})\cong Ell(\mathcal{J}_B)$ implies that $M_{loc}(A) \cong M_{loc}(B)$, and $M_{loc}(A)$ and $M_{loc}(B)$ are purely infinite. Furthermore, if $\mathcal{J}_A$ and $\mathcal{J}_B$ belong to $\mathcal{M}_l$, then

\begin{equation*} Ell(A) \cong Ell(B) \mbox{if and only if} \;\, A\cong B.\end{equation*}

Proof. Let $\mathcal{J}_{A}\cong \bigoplus_{i=1}^n \mathcal{J}^A_i$ and $\mathcal{J}_{B}\cong \bigoplus_{i=1}^n \mathcal{J}^B_i$ be G ₀-ideals in A and B. Then, all ideals $\mathcal{J}^A_i$ and $\mathcal{J}^B_i$ are Kirchberg algebras in the UCT class.

Recall that according to Zhang’s Dichotomy [49, Theorem 1.2(i)], every σ-unital, purely infinite and simple C*-algebra is either unital or stable. Furthermore, according to the Kirchberg–Phillips classification theorem [38, Theorem 8.4.1(ii)], the pair $(K_0(.), K_1(.))$ (resp. the triple $(K_0(.), [1]_0, K_1(.))$) is a complete invariant for stable (resp. unital) Kirchberg algebras in the UCT class.

Let n = 1. Then, $\mathcal{J}_{A}$ and $\mathcal{J}_{B}$ are Kirchberg algebras in the UCT class that are either stable or unital. Therefore (in both stable and unital cases), $Ell(\mathcal{J}_{A})\cong Ell(\mathcal{J}_B)$ if and only if $\mathcal{J}_{A} \cong \mathcal{J}_{B}$, and hence $Ell(\mathcal{J}_{A})\cong Ell(\mathcal{J}_B)$ implies that, $M_{loc}(A) \cong M_{loc}(\mathcal{J}_A) \cong M_{loc}(\mathcal{J}_B) \cong M_{loc}(B).$ Of course, $M_{loc}(A)$ and $M_{loc}(B)$ are purely infinite, by Lemma 2.18.

Let n > 1 and all $\mathcal{J}^A_i$ and $\mathcal{J}^B_i$ are also stable and with non-zero simple K_j-groups ($j=0,1$). Then, $Ell(\mathcal{J}_{A})\cong Ell(\mathcal{J}_B)$ implies that

\begin{equation*}\bigoplus_{i=1}^n K_j(\mathcal{J}^A_i) \cong \bigoplus_{i=1}^n K_j(\mathcal{J}^B_i),\end{equation*}

[45, Propositions 6.2.1 and 7.1.11(4)]. Since all $\mathcal{J}^A_i$ and $\mathcal{J}^B_i$ have non-zero simple K_j-groups ($j=0,1$), by changing the index, if necessary, $ K_j(\mathcal{J}^A_i) \cong K_j(\mathcal{J}^B_i)$, and hence $\mathcal{J}^A_i\cong \mathcal{J}^B_i$, for any $i\in \{1, 2, \ldots, n\}$. This means that,

\begin{equation*}M_{loc}(\mathcal{J}_A) \cong \bigoplus_{i=1}^n M_{loc} (\mathcal{J}^A_i) \cong \bigoplus_{i=1}^n M_{loc} (\mathcal{J}^B_i) \cong M_{loc}(\mathcal{J}_B),\end{equation*}

and hence $M_{loc}(A) \cong M_{loc}(B)$. Also, $M_{loc}(A)$ and $M_{loc}(B)$ are purely infinite, again according to Lemma 2.18.

For the last assertion (for every $n\geq 1$), since $ \mathcal{J}_A , \mathcal{J}_B \in \mathcal{M}_l$,

\begin{equation*}A \leq M_{loc}(A) \cong M_{loc}(\mathcal{J}_A ) = \mathcal{J}_A,\end{equation*}

and so $A= \mathcal{J}_A$. Similarly, $B= \mathcal{J}_B$. This implies that $ Ell(A) \cong Ell(B)$ if and only if $A\cong B$, by previous assertion.

Example 2.21. For every prime number p, let

be the graph introduced on p. 496 of [40]. The graph C*-algebra $C^*(\mathcal{F}_{p+1})$ is Artinian, prime, purely infinite [5, Proposition 5.3], separable and nuclear [3, p. 65]. The ideal corresponding to the saturated hereditary subset $H=\{w_1\}$ is I_H [3, Theorem 2.1.6(a)] that is minimal and essential. Since

\begin{equation*}I_H \sim_M C^*(E_H)\cong \mathcal{O}_{p+1}\end{equation*}

[3, Theorem 2.1.6(c)], the Brown–Green–Rieffel theorem [36, Theorem 5.55] implies that I_H and $C^*(E_H)$ are stably isomorphic. But the UCT class is closed under stable isomorphism [12, Remark 7.3]. Thus I_H is in the UCT class. Now, let

\begin{equation*}A:= C^*(\mathcal{F}_{p+1})\otimes \Bbb B.\end{equation*}

Then, the ideal $I_A= C^*(\mathcal{F}_{p+1})\otimes \Bbb K$ [29, Theorem 1.3] is an Artinian G-ideal and $\mathcal{J}_A= I_H \otimes \Bbb K$ is a simple G ₀-ideal in A. Thus, A is a non-separable and non-nuclear C*-algebra with Goldie dimension one and with infinitely many ideals. The ideal $\mathcal{J}_A$ is stable, separable, nuclear, purely infinite and in the UCT class and with simple K ₀-group

\begin{equation*}K_0(\mathcal{J}_A)\cong K_0(I_H)\cong K_0(\mathcal{O}_{p+1}) \cong \Bbb Z_p,\end{equation*}

and K ₁-group $K_1(\mathcal{J}_A)=\{0\}$ [37, Exercises 4.5 and 8.10]. Moreover, $M_{loc}(A)$ is also purely infinite.

Next, we study the decomposability of C*-algebras with Goldie dimension.

Definition 2.22. A C*-algebra A is said $\it{extending}$, if every essentially closed ideal in A is a direct summand.

Equivalently, a C*-algebra is extending if and only if for every $I\unlhd A$ there exists an ideal J in A such that it is a direct summand of A and $I\unlhd_e J$. Indeed, if A is an extending C*-algebra, $I\unlhd A$ and $\Omega=\{K \, | \, I\unlhd_e K\unlhd A \}$, then by Zorn’s Lemma, Ω has a maximal member J. Now, according to Definition 2.4 and maximality, J is essentially closed. Since A is extending, J is a direct summand. Converesly, let $I\unlhd_{e.c} A$. Then for $I\unlhd A$, there is a direct summand K such that $I\unlhd_e K$. Thus I = K, and so I is a direct summand.

Example 2.23. (a) For every $n \in \Bbb N^{\geq 2}$, the C*-algebra $C^*(\mathcal{G})$ (resp. $B= C^*(\mathcal{G})\oplus \Bbb B$) is extending and with Goldie dimension one (resp. two) that are separable and nuclear (resp. non-separable and non-nuclear) with infinitely many ideals. In general, a direct sum of simple (prime) C*-algebras is an extending C*-algebra.

(b) For $n_1, n_2\in \Bbb N^{\geq 2}$, consider the following graph:

The graph E ₁ is row-finite and satisfies Condition (K), and so there is a lattice isomorphism from the lattice of saturated hereditary subsets of E ₁ onto the lattice of ideals of $C^*(E_1)$, by [5, Theorem 4.1] and [6, Corollary 3.8]. The only non-empty saturated hereditary subsets of E ₁ are $H_1=\{w_1\}$, $H_2=\{w_2\}$ and $E_1^0$. Thus, $C^*(E_1)$ has only three non-zero ideals $I_{H_1}$, $I_{H_2}$ and $C^*(E_1)$ (and so it is Artinian and Noetherian). Clearly, $I_{H_1}$ (resp. $I_{H_2}$) is essentially closed and isomorphic with the Cuntz algebra $\mathcal{O}_{n_1}$ (resp. $\mathcal{O}_{n_2}$). Moreover, since the sum of two closed ideals in a C*-algebra is itself a closed ideal, we must have,

\begin{equation*}C^*(E_1)\cong I_{H_1}\oplus I_{H_2}.\end{equation*}

Of course, the only G-ideal (also K-ideal) in $C^*(E_1)$ is itself. Thus, $C^*(E_1)$ is a C*-algebra with Goldie dimension two (and with Krull dimension zero) which is extending but not prime. It is worth mentioning that the essentially closed ideals $I_{H_1}$ and $I_{H_2}$ are not essential ideals in $C^*(E_1)$.

Theorem 2.24. Let A be a C*-algebra. Then, the following conditions are equivalent:

(a) A has Goldie dimension.

(b) Every ascending (resp. descending) chain of essentially closed ideals of A stabilizes.

If, in addition, A is extending, then conditions (a) and (b) above are equivalent to:

(c) A is a finite direct sum of prime C*-algebras.

Proof. $(a \Rightarrow b)$: Let A has Goldie dimension and $J_1 \subsetneqq J_2 \subsetneqq J_3 \subsetneqq \ldots $ be a strictly ascending sequence of essentially closed ideals in A. Then for $i, j \in \Bbb N$, ($i \lneqq j$), we have $J_i \subsetneqq J_j$ and, in addition, $J_i \unlhd_{e.c} J_j$. Thus, J_i and $J_j/J_i$ have Goldie dimension and,

\begin{equation*}\mathrm{G.dim}(J_j) =\mathrm{G.dim}(J_i) + \mathrm{G.dim}(J_j/J_i),\end{equation*}

by Theorem 2.7(iv), and hence $\mathrm{G.dim}(J_i) \lneqq \mathrm{G.dim}(J_j)$. Thus, we have a strictly ascending sequence,

\begin{equation*}\mathrm{G.dim}(J_1) \lneqq \mathrm{G.dim}(J_2) \lneqq \mathrm{G.dim}(J_3) \lneqq \ldots,\end{equation*}

which is a contradiction. The descending case is proved similarly.

$(b \Rightarrow a)$: Let A does not have Goldie dimension. Thus, A contains an infinite direct sum of non-zero ideals, say $\bigoplus_{i=1}^{\infty} I_i$, (see Theorem 2.7(i)). We set $J_n := \bigoplus_{i=n}^\infty I_i$, ($n \in \Bbb N$).

First, we prove the ascending case. Since $I_1 \cap J_2 =\{0\}$, by Zorn’s Lemma, the set $\Omega=\{K \, | \, K\unlhd A \; \mbox{and} \; K \cap J_2 =\{0\} \}$ has a maximal member L ₁. Thus, $L_1 \unlhd_{e.c} A$, by maximality. Since $(L_1 \oplus I_2) \cap J_3 =\{0\}$, Similarly to the previous part, there is an essentially closed ideal $L_2 \supseteq L_1 \oplus I_2$. Also, since $(L_2 \oplus I_3) \cap J_4 =\{0\}$, there is an essentially closed ideal $L_3 \supseteq L_2 \oplus I_3$. By continuing this process, we will have a strictly ascending sequence $L_1 \subsetneqq L_2 \subsetneqq L_3 \subsetneqq \ldots $ of essentially closed ideals in A, which contradicts the assumption.

Now, we prove the assertion in the descending case. Let $K_1=A$. Since $I_1 \cap J_2 =\{0\}$, there is an ideal $K_2 \supseteq J_2$ such that $K_2 \unlhd_{e.c} K_1$. Also, since $I_2 \cap J_3 =\{0\}$, there is an ideal $K_3 \supseteq J_3$ such that $K_3 \unlhd_{e.c} K_2$. Note that the relation $\unlhd_{e.c} $ is transitive, by Lemma 2.6, and hence K ₃ is also essentially closed in $K_1=A$. By continuing this process, we get a strictly descending sequence $L_1 \supsetneqq L_2 \supsetneqq L_3 \supsetneqq \ldots $ of essentially closed ideals in A, which is a contradiction.

$(c \Rightarrow a)$: Let A be an extending C*-algebra and $A= \bigoplus_{i=1}^n A_i$, where each A_i is a prime C*-algebra and $n \in \Bbb N$. Then,

\begin{equation*} \mathrm{G.dim}(A)= \sum_{i=1}^n \mathrm{G.dim}(A_i)=n.\end{equation*}

$(a \Rightarrow c)$: Let A be an extending C*-algebra with Goldie dimension. Thus, A contains an ideal I ₁ that is a prime C*-algebra, and there are an ideal J ₁ and a C*-subalgebra B ₁ of A such that $A= J_1\oplus B_1$ and $I_1 \unlhd_e J_1$. Then B ₁ is also extending. Because if L ₁ is an essentially closed ideal in B ₁, then since B ₁ is a direct summand of A (and so is essetially closed in A), L ₁ is essentially closed in A, by transitivity. But A is extending, and hence L ₁ is a direct summand of A (and so of B ₁). Furthermore, J ₁ is a prime C*-algebra ($\mathrm{G.dim}(J_1)= \mathrm{G.dim}(I_1)=1$). Now, since $\mathrm{G.dim}(B_1) \leq \mathrm{G.dim}(A)$, B ₁ has also Goldie dimension. By repeating the previous process after a finite number of times, the assertion is obtained.

Example 2.25. For $n,n_1, n_2 \in \Bbb N^{\geq 2}$, consider the following graph E ₂:

The only non-empty saturated hereditary subsets of E ₂ are $H_1=\{w_1\}$, $H_2=\{w_2\}$, $H_3=\{w_1, w_2\}$ and $E_2^0$. The graph C*-algebra $C^*(E_2)$ has Goldie dimension two, but it cannot be written as a finite direct sum of prime C*-algebras, and hence is not extending, by Theorem 2.24. In fact, its only G-ideal in $C^*(E_2)$ is $I_{H_3}(\cong I_{H_1} \oplus I_{H_2})$, which is the direct sum of two prime C*-algebras.

A Hilbert C*-module X over a C*-algebra A (or a Hilbert A-module) is a vector space over $\Bbb C$ and a right A-module with a compatible A-valued inner product $\langle \ldots \rangle_A$ (see [36, Definition 2.8], for a detailed definition). A Hilbert A-module X is called full if $\langle X , X \rangle_A=A$. Let $\mathcal{L}_A(X)$ be the C*-algebra of adjointable operators on X and $\mathcal{K}_A(X)$ the closed linear span of the set of all elementary operators $\theta_{x,y} : X \to X $ given by $\theta_{x,y} (z)= x \langle y , z \rangle; x,y,z\in X$ [26, 36]. For an ideal I in A, an ideal submodule for X, denoted by X_I, is defined by,

\begin{equation*}X_I:=\overline{span} \{x.a \;:\; x\in X \;, \; a\in I\}=\{x.a \;:\; x\in X \;, \; a\in I\},\end{equation*}

(see [4, Definition 1.1 and Proposition 1.2]). Let $\pi: A \to A/I$ and $q:X \to X/X_I$ be the quotient maps. Then, $X/X_I$ with the right action $q(x)\pi(a)=q(xa)$ and the inner product $\langle q(x) , q(y) \rangle=\pi(\langle x , y \rangle)$ is a Hilbert $A/I$-module [4].

If $A\subseteq B$ is an inclusion of C*-algebras with a common identity and a faithful canonical conditional expectation $\mathcal{E} : B \to A$, and

\begin{equation*}\|x\|_{\mathcal{E}}^2=\| \mathcal{E}(x^*x) \|\end{equation*}

is the norm induced by the inner product $\langle x , y \rangle_A= \mathcal{E}(x^*y)$, then, $X_{\mathcal{E}}:=\overline{B}^{\| . \|_{\mathcal{E}} }$ is a Hilbert A-module. A conditional expectation $\mathcal{E} : B \to A$ is said to have finite index if there exists $ z_1 , z_2 , \ldots, z_n\in B$ such that $ x = \sum_{i=1}^n z_i \mathcal{E}(z_i^*x)$, $x\in B $ [22, 44].

Theorem 2.26. (i) Every Hilbert C*-module X over a C*-algebra A with Goldie dimension n that is extending or an AW*-algebra is decomposed into a direct sum of n ideal submodules.

(ii) Let $A\subseteq B$ be an inclusion of C*-algebras with a common identity and a conditional expectation $ \mathcal{E}: B \to A$ that is faithful and of finite index. Then, A has Goldie dimension if and only if $\mathcal{K}_A(X_{\mathcal{E}})(= \mathcal{L}_A(X_{\mathcal{E}}))$ has Goldie dimension.

Proof. (i) First we show that if X is a Hilbert C*-module over a C*-algebra A, I and J are two closed ideals in A, and X_I and X_J be the associated ideal submodules, then

\begin{equation*} X_I+X_J= X_{I+J}\;\; \mbox{and}\; \; X_I \cap X_J = X_{I\cap J}.\end{equation*}

Since $X_I + X_J$ is closed in X [43, Lemma 2.12] and, for every ideal K in A, $X_K=\{x.k \;:\; x\in X \;, \; k\in K\}$, we have $X_I+X_J= X_{I+J}$. Note that every ideal submodule in an ideal submodule of X is also an ideal submodule in X. If $\pi_0: I\to (I+J)/J$ is the restriction map of the quotient map

\begin{equation*}\pi: I+J\to (I+J)/J,\end{equation*}

then $\mathrm{Ker}(\pi_0)=I\cap J$. Let $q_0: X_I \to (X_I + X_J)/X_J$ be the restriction map of the quotient map

\begin{equation*}q: X_I + X_J \to (X_I + X_J)/X_J.\end{equation*}

Then q ₀ is a π ₀-morphism (see [4, Definition 2.1]), since

\begin{equation*}\langle q_0(x) , q_0(y) \rangle= \langle q(x) , q(y) \rangle= \pi (\langle x , y \rangle)= \pi_0(\langle x , y \rangle),\end{equation*}

for all $x, y \in X_I$. Thus, [4, Theorem 2.3] implies that,

\begin{equation*}X_I \cap X_J = \mathrm{Ker}(q_0)= X_{\mathrm{Ker}(\pi_0)}= X_{I\cap J}.\end{equation*}

Now, let A be a C*-algebra with Goldie dimension n that is extending or is an AW*-algebra. If A is extending, then it is a direct sum of n prime C*-algebras, by Theorem 2.24 and Lemma 2.1. If A is an AW*-algebra and $I_A=\bigoplus_{i=1}^n I_i$ a G-ideal in A, then

\begin{equation*}A=M_{loc}(A)=\bigoplus_{i=1}^n M_{loc}(I_i),\end{equation*}

where every $M_{loc}(I_i)$ is a prime C*-algebra [2, Proposition 3.3.2]. In both cases, there are prime C*-algebras $A_1,A_2, \ldots, A_n$ in A such that $A=\bigoplus_{i=1}^n A_i$. Thus,

\begin{equation*}X_A= X_{\bigoplus_{i=1}^n A_i }=\bigoplus_{i=1}^n X_{A_i}.\end{equation*}

(ii) First note that since $\mathcal{E}$ has finite index, there exists $ z_1 , z_2 , \ldots, z_n\in B$ such that,

\begin{equation*}x=\sum_{i=1}^n z_i \mathcal{E}(z_i^*x) = \sum_{i=1}^n \theta_{z_i,z_i}(x); \;x\in B.\end{equation*}

Since B is $\| . \|_{\mathcal{E}}$-dense in $X_\mathcal{E}$, $ \mathcal{K}(X_\mathcal{E})$ is unital, with $1_{\mathcal{K}(X_\mathcal{E})}=\sum_{i=1}^n \theta_{z_i,z_i},$ and hence $\mathcal{K}_A(X_{\mathcal{E}})=M(\mathcal{K}_A(X_{\mathcal{E}}))=\mathcal{L}_A(X_{\mathcal{E}}).$ Furthermore, observe that $X_{\mathcal{E}}$ is full. Indeed, the elements in a C*-algebra A can be expressed as linear combinations of positive elements, and every positive element is in the form $a^*a(=\mathcal{E}(a^*a)=\langle a , a\rangle_A)$, for an $a\in A$. Thus,

\begin{equation*} A\subseteq \overline{span} \{\mathcal{E}(x^*y) \, | \, x,y\in X_{\mathcal{E}}\}= \langle X_{\mathcal{E}} , X_{\mathcal{E}} \rangle_A.\end{equation*}

Now, since every full Hilbert A-module X is a $\mathcal{K}_A(X)$-A-imprimitivity bimodule [36, Proposition 3.8], we have $\mathcal{K}_A(X_{\mathcal{E}}) \sim_M A.$ Therefore, the assertion holds, by Theorem 2.7(iii).

Theorem 2.26(ii) shows that Corollary 2.2.14 of [44] (if A is simple, then $\mathcal{K}_A(X_\mathcal{E})$ is simple), is also true for C*-algebras with Goldie dimension. Note that $\mathcal{K}_A(X_\mathcal{E})$ is exactly $C^*\langle B,e_A\rangle$, by the remark on p. 40, Definition 2.2.10 and Lemma 2.2.9 of [44].