1. Introduction and Preliminaries
Kadison and Kastler in  initiated the study of uniform perturbations of operator algebras. They considered a fixed C*-algebra and equipped the set of all C*-subalgebras of with a metric arising from Hausdorff distance between the unit balls of these subalgebras. We first recall the following definition of the metric d defined on the set of all C*-subalgebras of a C*-algebra (see  ).
Definition 1.1. Let and be C*-subalgebras of a C*-algebra . The Kadison-Kastler metric between and is defined by
where and denote the unit ball of and respectively.
Kadison and Kastler conjectured in  that sufficiently close von Neumann algebras (or C*-algebras) are necessarily unitarily conjugate. The first positive answer to Kadison-Kastler’s conjecture was given by Christensen  when either or is a von Neumann algebra of type I. Many results related to this conjecture have been obtained during the past 40 years (     ). One-sided version of Kadison-Kastler’s conjecture was introduced and studied by Christensen in  as well. Christensen showed in  that a nuclear C*-algebra that is nearly contained in an injective von Neumann algebra is unitarily conjugate to this von Neumann algebra. Christensen, Sinclair, Smith and White showed in  that the property of having a positive answer to Kadison’s similarity problem transfers to close C*-algebras. Very recently, Kadison-Kastler’s conjecture has been proved for the class of separable nuclear C*-algebras in the remarkable paper  .
The problem we are going to consider is as follows: Suppose are C*-subalgebras of a C*-algebra . If , is and share similar properties?
In this short note, we show that the sets of matricial field algebras (MF algebras) and quasidiagonal C*-algebras of a given C*-algebra are closed under the perturbation of uniform norm.
2. Main Results
In this section, we consider some topological properties of the set of all MF algebras and quasidiagonal C*-subalgebras under the perturbation of uniform norm. For basics of C*-algebras, we refer to  and  . We first recall the definition of MF algebras (  ).
Suppose is a sequence of complex matrix algebras. We can introduce the full C*-direct product of as follows:
Furthermore, we can introduce a norm closed two sided ideal in as follows,
Let be the quotient map from to . It is known that is a unital C*-algebra. If we denote by , then
Now we are ready to recall an equivalent definition of MF algebras which is given by Blackadar and Kirchberg (  ).
Definition 2.1. (Theorem 3.2.2,  ) Let be a separable C*-algebra. If can be embedded as a C*-subalgebra of for a sequence of integers, then is called an MF algebra.
Lemma 2.2. (  Lemma 2.12) Suppose that is a separable C*-algebra. Assume for every finite family of elements in and every , there is an MF algebra such that , (in the sense of Definition 2.3 in  ). Then is also an MF algebra.
Proposition 2.3. Let be a C*-algebra and be the subset of all separable MF algebras contained in . Then is closed under the metric d.
Proof. Let . Then there exist such that . For any , , there is an such that . Then there exist such that
for all i. It follows from Lemma 2.2 that is also a MF algebra.+
We will recall some results about quasidiagonal C*-algebras for the reader’s convenience. We refer the reader to  for a comprehensive treatment of this important class of C*-algebras.
Definition 2.4. A subset is called a quasidiagonal set of operators if for each finite set , finite set and , there exists a finite rank projection such that and for all and .
Definition 2.5. A C*-algebra is called quasidiagonal (QD) if there exists a faithful representation such that is a quasidiagonal set of operators.
The following result is Lemma 7.1.3 in  which is useful to determine whether a C*-algebra is quasidiagonal or not.
Lemma 2.6. A C*-algebra is quasidiagonal if and only if for each finite set and , there exists a completely positive map such that
for all .
Proposition 2.7. Let be a separable C*-algebra. Let be the set of all quasidiagonal C*-subalgebras of . Then is closed under the metric d.
Proof. Let and choose such that . Given finite subset of the unit ball of and . There is a
such that . Choose in the unit ball of such that for . Since is QD, it follows from Lemma 2.6 that there is a c.c.p. map such that
for all . Now use Arveson’s extension theorem (  ) to extend to a c.c.p. map from to . Let be the restriction of to . Then for each , we have
Use Lemma 2.6 again we have that is quasidiagonal.+
In this paper, we use some characterizations of MF algebras and quasidiagonal C*-algebras to show that these two sets of C*-subalgebras of a given C*-algebras are closed with respect to the topology induced by the Kadison-Kastler metric.
Partially supported by NSFC (11871303 and 11671133) and NSF of Shandong Province (ZR2019MA039).
 Christensen, E., Sinclair, A.M., Smith, R.R. and White, S.A. (2010) Perturbations of C*-Algebraic Invariants. Geometric and Functional Analysis, 20, 368-397.
 Brown, N.P. and Ozawa, N. (2008) C-Algebras and Finite-Dimensional Approximations, Graduate Studies in Mathematics. Vol. 88, American Mathematical Society, Providence.