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Perturbation problem of operator algebras was first introduced by Kadison and Kastler. In this short note, we consider the uniform perturbation of two classes of operator algebras,
i.e., MF algebras and quasidiagonal C
^{*}-algebras. We show that the sets of MF algebras and quasidiagonal C
^{*}-algebras of a given C
^{*}-algebra are closed under the perturbation of uniform norm.

Kadison and Kastler in [^{*}-algebra U and equipped the set of all C^{*}-subalgebras of U 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 U (see [

Definition 1.1. Let A and B be C^{*}-subalgebras of a C^{*}-algebra U . The Kadison-Kastler metric d ( A , B ) between A and B is defined by

d ( A , B ) = max { sup a ∈ ( A ) 1 inf b ∈ ( B ) 1 ‖ a − b ‖ , sup b ∈ ( B ) 1 inf a ∈ ( A ) 1 ‖ a − b ‖ }

where ( A ) 1 and ( B ) 1 denote the unit ball of A and B respectively.

Kadison and Kastler conjectured in [^{*}-algebras) are necessarily unitarily conjugate. The first positive answer to Kadison-Kastler’s conjecture was given by Christensen [^{*}-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 [^{*}-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 A , B are C^{*}-subalgebras of a C^{*}-algebra U . If d ( A , B ) < γ , is A and B 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.

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 [

Suppose { M k n ( ℂ ) } n = 1 ∞ is a sequence of complex matrix algebras. We can introduce the full C*-direct product ∏ m = 1 ∞ M k m ( ℂ ) of { M k n ( ℂ ) } n = 1 ∞ as follows:

∏ n = 1 ∞ M k n ( ℂ ) = { ( Y n ) n = 1 ∞ | ∀ n ≥ 1, Y n ∈ M k n ( C ) and sup n ≥ 1 ‖ Y n ‖ < ∞ } . (1)

Furthermore, we can introduce a norm closed two sided ideal in ∏ n = 1 ∞ M k n ( ℂ ) as follows,

∑ n = 1 ∞ M k n ( ℂ ) = { ( Y n ) n = 1 ∞ ∈ ∏ n = 1 ∞ M k n ( ℂ ) : lim n → ∞ ‖ Y n ‖ = 0 } . (2)

Let π be the quotient map from ∏ n = 1 ∞ M k n ( ℂ ) to ∏ n = 1 ∞ M k n ( ℂ ) / ∑ n = 1 ∞ M k n ( ℂ ) . It is known that ∏ n = 1 ∞ M k n ( ℂ ) / ∑ n = 1 ∞ M k n ( ℂ ) is a unital C*-algebra. If we denote π ( ( Y n ) n = 1 ∞ ) by [ ( Y n ) n ] , then

‖ [ ( Y n ) n ] ‖ = lim sup n → 1 ‖ Y n ‖ . (3)

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, [^{*}-algebra. If U can be embedded as a C^{*}-subalgebra of ∏ n = 1 ∞ M k n ( ℂ ) / ∑ n = 1 ∞ M k n ( ℂ ) for a sequence { k n } n = 1 of integers, then U is called an MF algebra.

Lemma 2.2. ( [^{*}-algebra. Assume for every finite family of elements x 1 , x 2 , ⋯ , x n in U and every ε > 0 , there is an MF algebra U 1 such that { x 1 , x 2 , ⋯ , x n } ⊂ ε U 1 , (in the sense of Definition 2.3 in [

Proposition 2.3. Let U be a C^{*}-algebra and F be the subset of all separable MF algebras contained in U . Then F is closed under the metric d.

Proof. Let A ∈ F ¯ . Then there exist A n ∈ F such that d ( A n , A ) → 0 . For any x 1 , x 2 , ⋯ , x m ∈ A , ∀ ε > 0 , there is an n 0 such that d ( A n 0 , A ) < ε 2 ∑ i = 1 m ‖ x i ‖ + 1 . Then there exist y 1 , y 2 , ⋯ , y m ∈ A n 0 such that

‖ x i − y i ‖ < ε 2 ∑ i = 1 m ‖ x i ‖ + 1 ‖ x i ‖ < ε (4)

for all i. It follows from Lemma 2.2 that A is also a MF algebra.+

We will recall some results about quasidiagonal C^{*}-algebras for the reader’s convenience. We refer the reader to [^{*}-algebras.

Definition 2.4. A subset Ω ⊂ B ( H ) is called a quasidiagonal set of operators if for each finite set ω ⊂ Ω , finite set χ ⊂ H and ε > 0 , there exists a finite rank projection P ∈ B ( H ) such that ‖ T P − P T ‖ ≤ ε and ‖ P ( x ) − x ‖ ≤ ε for all T ∈ ω and x ∈ χ .

Definition 2.5. A C^{*}-algebra U is called quasidiagonal (QD) if there exists a faithful representation π : U → B ( H ) such that π ( U ) is a quasidiagonal set of operators.

The following result is Lemma 7.1.3 in [^{*}-algebra is quasidiagonal or not.

Lemma 2.6. A C^{*}-algebra U is quasidiagonal if and only if for each finite set F ⊂ U and ε > 0 , there exists a completely positive map ϕ : U → M n ( ℂ ) such that

‖ ϕ ( a b ) − ϕ ( a ) ϕ ( b ) ‖ < ϵ (5)

and

‖ ϕ ( a ) ‖ > ‖ a ‖ − ϵ (6)

for all a , b ∈ F .

Proposition 2.7. Let U be a separable C^{*}-algebra. Let F = Q D ( U ) be the set of all quasidiagonal C^{*}-subalgebras of U . Then F is closed under the metric d.

Proof. Let A ∈ F ¯ and choose A n ∈ F such that d ( A n , A ) → 0 . Given finite subset { x 1 , x 2 , ⋯ , x k } of the unit ball of A and ϵ > 0 . There is a

N ∈ ℕ such that d ( A N , A ) < ϵ 6 . Choose y 1 , y 2 , ⋯ y k in the unit ball of A N such that ‖ x i − y i ‖ < ϵ 6 for i = 1 , 2 , ⋯ , k . Since A N is QD, it follows from Lemma 2.6 that there is a c.c.p. map ϕ : A N → M t ( ℂ ) such that

‖ ϕ ( y i y j ) − ϕ ( y i ) ϕ ( y j ) ‖ ≤ ε 6 (7)

and

‖ ϕ ( y j ) ‖ ≥ ‖ y j ‖ − ϵ 6 (8)

for all i , j = 1 , 2 , ⋯ , k . Now use Arveson’s extension theorem ( [

‖ ψ ( x i x j ) − ψ ( x i ) ψ ( x j ) ‖ ≤ ‖ ψ ( x i x j ) − ψ ( x i y j ) ‖ + ‖ ψ ( x i y j ) − ψ ( y i y j ) ‖ + ‖ ψ ( y i y j ) − ψ ( y i ) ψ ( y j ) ‖ + ‖ ψ ( y i ) ψ ( y j ) − ψ ( x i ) ψ ( y j ) ‖ + ‖ ψ ( x i ) ψ ( y j ) − ψ ( x i ) ψ ( x j ) ‖ < ε (9)

and

‖ ψ ( x i ) ‖ = ‖ ψ ( x i − y i ) + ψ ( y i ) ‖ ≥ ‖ ψ ( y i ) ‖ − ε 6 > ‖ y i ‖ − ε 3 ≥ ‖ x i ‖ − ε . (10)

Use Lemma 2.6 again we have that A 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).

The authors declare no conflicts of interest regarding the publication of this paper.

Zhan, W.J. and Wang, L.G. (2019) A Note on the Perturbation of MF Algebras and Quasidiagonal C^{*}-Algebras. Journal of Applied Mathematics and Physics, 7, 2026-2030. https://doi.org/10.4236/jamp.2019.79139