1 Introduction

In 1955, Kadison in his article [8] introduced the following problem which still remains one of the most famous open problems in the theory of \(C^*\)-algebras:

Let H be a Hilbert space and \(u: {\mathcal {A}}\rightarrow {\mathcal {B}}(H)\) be a bounded homomorphism on a \(C^*\)-algebra \({\mathcal {A}}.\) Does there exist an invertible operator \(S\in {\mathcal {B}}(H)\) such that the map

$$\begin{aligned} \pi : {\mathcal {A}}\rightarrow {\mathcal {B}}(H), \,\,A\mapsto \pi (A)=S^{-1} u(A) S \end{aligned}$$

defines a \(*\)-homomorphism?

We say that a \(C^*\)-algebra \({\mathcal {A}}\) satisfies the similarity property ((SP)) if every bounded homomorphism \(u: {\mathcal {A}}\rightarrow {\mathcal {B}}(H),\) where H is a Hilbert space, has the above property. In [6], Haagerup gave an affirmative answer for the case in which the map u has a finite cyclic set, i.e. there exist \(n\in {\mathbb {N}}\) and \(\xi _1,\ldots , \xi _n\in H\) such that

$$ H=\overline{[u(A)\xi _i\,\mid \,\,A\in {\mathcal {A}},\,\,i=1,\ldots ,n]}, $$

where, in general, for a subset \({\mathcal {S}}\) of some vector space \({\mathcal {X}},\) \(\,[{\mathcal {S}}]\) denotes the linear span of \({\mathcal {S}}.\) In the same article, Haagerup proved that every completely bounded homomorphism from a \(C^*\)-algebra \({\mathcal {A}}\) into \({\mathcal {B}}(H),\) where H is a Hilbert space, is similar to a \(*\)-homomorphism.

Kadison’s similarity problem is equivalent to Arveson’s hyperreflexivity problem: Every von Neumann algebra is hyperreflexive. Arveson in [1] showed that all nest algebras are hyperreflexive with hyperreflexivity constant 1. Although many von Neumann algebras are hyperreflexive, the question of whether every von Neumann algebra is hyperreflexive remains open.

We say that a von Neumann algebra \({\mathcal {M}}\) satisfies the weak similarity property ((WSP)) if every w*-continuous, unital and bounded homomorphism \(u: {\mathcal {M}}\rightarrow {\mathcal {B}}(H),\) where H is a Hilbert space, is similar to a \(*\)-homomorphism. The connection between (SP) and (WSP) is given by the following lemma. Its proof can be found in [5, Lemma 1.1].

Lemma 1.1

Let \({\mathcal {A}}\) be a unital \(C^*\)-algebra. The algebra \({\mathcal {A}}\) satisfies (SP) if and only if the second dual algebra \({\mathcal {A}}^{**}\) satisfies (WSP).

Recently, the author and Eleftherakis in their joint work [5] proved the following theorem.

Theorem 1.2

A von Neumann algebra \({\mathcal {M}}\) satisfies (WSP) if and only if the von Neumann algebras \({\mathcal {M}}^{\prime }\bar{\otimes } {\mathcal {B}}(\ell ^2(I))\) are hyperreflexive for all cardinals I. Here, \({\mathcal {M}}^{\prime }\) is the commutant of \({\mathcal {M}}.\)

In the special case of a separably acting von Neumann algebra \({\mathcal {M}},\) we proved the following corollary.

Corollary 1.3

Let \({\mathcal {M}}\subseteq {\mathcal {B}}(H)\) be a von Neumann algebra and H be a separable Hilbert space. Then \({\mathcal {M}}\) satisfies (WSP) if and only if the algebra \({\mathcal {M}}^\prime \bar{\otimes } {\mathcal {B}}(\ell ^2({\mathbb {N}}))\) is hyperreflexive.

In general, a \(\textrm{w}^*\)-closed space \({\mathcal {X}}\) is called completely hyperreflexive if the space \({\mathcal {X}}\bar{\otimes } {\mathcal {B}}(\ell ^2({\mathbb {N}}))\) is hyperreflexive. It is obvious that completely hyperreflexive spaces are hyperreflexive. On the other hand, whether hyperreflexivity implies complete hyperreflexivity remains an open problem [3, 4].

In the same article [5], the author and Eleftherakis introduced the following hypothesis (CHH): Every hyperreflexive separably acting von Neumann algebra is completely hyperreflexive.

As we showed in Section 3 of our article [5], the resolution of this problem yields an affirmative answer to the similarity problem.

Throughout this paper, if \({\mathcal {M}}\) and \({\mathcal {N}}\) are von Neumann algebras, then we denote by \({\mathcal {M}}\bar{\otimes } {\mathcal {N}}\) their spatial tensor product. We introduce the following hypothesis (EP): Every separably acting von Neumann algebra with a cyclic vector is hyperreflexive.

The hypothesis we introduce is weaker than Arveson’s. Arveson conjectured that all von Neumann algebras are hyperreflexive, while we restrict on the ones that are separably acting and have a cyclic vector.

In Sect. 2, we set the stage by presenting useful definitions, we recall lemmas and theorems that were proven in [5] and are useful for the next section.

In Sect. 3, we prove that under (EP), every separably acting von Neumann algebra satisfies (WSP). Using this result, we show that under (EP), all \(C^*\)-algebras satisfy (SP).

In what follows, \(\,{\mathcal {B}}(H,K)\) is the Banach space of all linear and bounded operators from the Hilbert space H to the Hilbert space K. If \(H=K,\) then we simply write \({\mathcal {B}}(H,H)={\mathcal {B}}(H)\) for the von Neumann algebra of all linear and bounded operators from the Hilbert space H to itself. Furthermore, if \({\mathcal {D}}\) is a set of operators acting on H then

$${\mathcal {D}}^\prime =\left\{ T\in {\mathcal {B}}(H)\mid T D=D\,T,\,\,\forall \,D\in {\mathcal {D}}\right\} $$

is the commutant of \({\mathcal {D}}.\)

2 Preliminaries

In this section we recall some basic definitions and facts that will be used throughout the article.

Let \({\mathcal {M}}\) be a von Neumann algebra acting on the Hilbert space H. If \(T\in {\mathcal {B}}(H)\) we set

$$\begin{aligned} d(T,\,{\mathcal {M}}) =\inf \left\{ \Vert T-X\Vert \,\,\mid X\in {\mathcal {M}}\right\} \end{aligned}$$

to be the distance from T to \({\mathcal {M}}\). We also set

$$\begin{aligned} r_{{\mathcal {M}}}(T) =\sup _{\Vert \xi \Vert =\Vert \eta \Vert =1} \left\{ \left| \left\langle {T\xi ,\eta }\right\rangle \right| \,\,\,\,\mid \left\langle {X\xi , \eta }\right\rangle =0,\,\forall \,X\in {\mathcal {M}}\right\} \end{aligned}$$

and it is easy to prove that \(r_{{\mathcal {M}}}(T)\le d(T,\,{\mathcal {M}}).\) Furthermore, it is immediate that the quantities \(d(\cdot ,\,{\mathcal {M}})\) and \(r_{{\mathcal {M}}}(\cdot )\) define seminorms on \({\mathcal {B}}(H)\) and the quotient space \({\mathcal {B}}(H)/{\mathcal {M}}\) is a Banach space with respect to the norm

$$\begin{aligned} ||T+{\mathcal {M}}||_{1}=d(T,\,{\mathcal {M}}),\,\,T\in {\mathcal {B}}(H). \end{aligned}$$

Lemma 2.1

If \(\,T\notin {\mathcal {M}},\) then \(r_{{\mathcal {M}}}(T)\ne 0.\)

The above lemma states that the quantity

$$\begin{aligned} ||T+{\mathcal {M}}||_{2}=r_{{\mathcal {M}}}(T),\,\,T\in {\mathcal {B}}(H) \end{aligned}$$

defines a norm on \({\mathcal {B}}(H)/{\mathcal {M}}\) but the quotient space \({\mathcal {B}}(H)/{\mathcal {M}}\) is not necessarily a Banach space with respect to this norm.

Definition 2.2

If there exists \(0<k<\infty \) such that

$$\begin{aligned} d(T,\,{\mathcal {M}}) \le k\,r_{{\mathcal {M}}}(T), \,\forall \,T\in {\mathcal {B}}(H), \end{aligned}$$

then we say that the von Neumann algebra \({\mathcal {M}}\) is hyperreflexive.

We observe that if \({\mathcal {M}}\) is hyperreflexive, then the quotient space \({\mathcal {B}}(H)/{\mathcal {M}}\) is a Banach space with respect to the norm \(||\cdot ||_{2}\) since in that case, the norms \(||\cdot ||_{1}\) and \(||\cdot ||_{2}\) are equivalent.

Lemma 2.1 and Definition 2.2 lead to the following definition.

Definition 2.3

Let \({\mathcal {M}}\) be a von Neumann algebra acting on the Hilbert space H. We define

$$\begin{aligned} k({\mathcal {M}})=\sup _{T\not \in \, {\mathcal {M}}} \frac{d(T,\, {\mathcal {M}}) }{r_{{\mathcal {M}}}(T)} \end{aligned}$$

to be the hyperreflexivity constant of \({\mathcal {M}}\).

Therefore, \({\mathcal {M}}\) is hyperreflexive if and only if \(k({\mathcal {M}})<\infty .\) Moreover, if \({\mathcal {M}}\) is hyperreflexive, then \(k({\mathcal {M}})\ge 1\) (since \(0<r_{{\mathcal {M}}}(T)\le d(T,\,{\mathcal {M}})).\)

In [5], the author and Eleftherakis, using arguments from Pisier’s book [10], proved some results connecting the weak similarity property with the notion of hyperreflexivity of von Neumann algebras. The most important of them are the following:

Theorem 2.4

[5] A von Neumann algebra \({\mathcal {M}}\) satisfies (WSP) if and only if the von Neumann algebras \({\mathcal {M}}^{\prime }\bar{\otimes } {\mathcal {B}}(\ell ^2(I))\) are hyperreflexive for all cardinals I.

Corollary 2.5

[5] Let \({\mathcal {M}}\subseteq {\mathcal {B}}(H)\) be a von Neumann algebra and H be a separable Hilbert space. Then \({\mathcal {M}}\) satisfies (WSP) if and only if the algebra \({\mathcal {M}}^\prime \bar{\otimes } {\mathcal {B}}(\ell ^2({\mathbb {N}}))\) is hyperreflexive.

Concluding this section, we remind the reader the definition of a completely bounded map between von Neumann algebras.

Definition 2.6

Let \({\mathcal {M}}\) and \({\mathcal {N}}\) be von Neumann algebras acting on the Hilbert spaces H and K respectively, and \(u: {\mathcal {M}}\rightarrow {\mathcal {N}}\) be a linear map. For each \(n\in {\mathbb {N}}\) consider the matrix amplification

$$\begin{aligned} u_n: _n({\mathcal {M}})\rightarrow M_n({\mathcal {N}}), \,\, u_n((X_{i,j}))=(u(X_{i,j})). \end{aligned}$$

We say that u is completely bounded if

$$\begin{aligned} ||u||_{cb}:=\sup _{n} ||u_n||<\infty . \end{aligned}$$

3 (EP) implies that all C*-algebras satisfy (SP)

In this section we prove that under (EP), every separably acting von Neumann algebra satisfies (WSP). As we will see, Corollary 2.5 plays a crucial role for the proof of this result. Using this result, we show that under (EP), all \(C^*\)-algebras satisfy (SP). There are a lot of separably acting von Neumann algebras with a cyclic vector which are hyperreflexive. Here are some examples:

Example 3.1

  1. (i)

    \({\mathcal {B}}(\ell ^2({\mathbb {N}}))\) is hyperreflexive with hyperreflexivity constant \(k({\mathcal {B}}(\ell ^2({\mathbb {N}})))=1\).

  2. (ii)

    If \({\mathcal {M}}\) is a separably acting abelian von Neumann algebra, then its commutant \({\mathcal {M}}^\prime \) has a cyclic vector and it is hyperreflexive.

  3. (iii)

    If \({\mathcal {M}}\) is a separably acting von Neumann algebra which is in standard form, then by [7, Lemma 2.8] it has a cyclic and a separating vector and then both \({\mathcal {M}}\) and \({\mathcal {M}}^\prime \) are hyperreflexive according to [9, Corollary 3.4].

  4. (iv)

    If \({\mathcal {M}}\) is a separably acting type \(II_{\infty }\) factor with commutant \({\mathcal {M}}^\prime \) of the same type, then \({\mathcal {M}}\) is hyperreflexive. If \({\mathcal {M}}^\prime \) is of type \(II_{1},\) is the algebra \({\mathcal {M}}\) hyperreflexive?

Remark 3.2

More specifically, if the second part of the example (iv) is true then the commutant \({\mathcal {M}}^\prime \) satisfies (WSP). Indeed, if \(k({\mathcal {M}})<\infty \) then,

$$\begin{aligned} k({\mathcal {M}}\bar{\otimes } {\mathcal {B}}(\ell ^2({\mathbb {N}})))=k({\mathcal {M}})<\infty \end{aligned}$$

since \({\mathcal {M}}\bar{\otimes } {\mathcal {B}}(\ell ^2({\mathbb {N}}))\) and \({\mathcal {M}}\) are unitarily equivalent, see [2, Examples (ii) III.1.5.6]. Thus, by Corollary 2.5 the commutant \({\mathcal {M}}^\prime \) satisfies (WSP).

Let us now return to our initial goal, that is, to prove that under (EP), all \(C^*\)-algebras satisfy (SP). First, we prove some useful lemmas.

Lemma 3.3

Assume that (EP) is true. Then every separably acting von Neumann with a separating vector satisfies (WSP).

Proof

Let \({\mathcal {M}}\) be a separably acting von Neumann algebra with a separating vector. Then the separably acting von Neumann algebra \({\mathcal {M}}^\prime \bar{\otimes } {\mathcal {B}}(\ell ^2 ({\mathbb {N}}))\) has a cyclic vector and under our hypothesis, the algebra \({\mathcal {M}}^\prime \bar{\otimes } {\mathcal {B}}(\ell ^2({\mathbb {N}}))\) is hyperreflexive. According to Corollary 2.5 we deduce that the von Neumann algebra \({\mathcal {M}}\) satisfies (WSP). \(\square \)

Lemma 3.4

Assume that (EP) is true. Then every separably acting von Neumann algebra satisfies (WSP).

Proof

Let \({\mathcal {M}}\) be a von Neumann algebra acting on the separable Hilbert space H. By [7, Theorem 1.6] the algebra \({\mathcal {M}}\) is isomorphic to a von Neumann algebra \({\mathcal {N}}\) which is standard on some Hilbert space K. Since H is separable, the von Neumann algebra \({\mathcal {M}}\) is \(\sigma \)-finite. Therefore, \({\mathcal {N}}\) is \(\sigma \)-finite and by [7, Lemma 2.8] the algebra \({\mathcal {N}}\) admits a cyclic and a separating vector. Lemma 3.3 yields that \({\mathcal {N}}\) satisfies (WSP), and thus \({\mathcal {M}}\) satisfies (WSP). \(\square \)

Lemmas 3.5 and 3.6 were proven in [5]. We present them here, as well, for the sake of completeness.

Lemma 3.5

[5] Let \({\mathcal {X}}\) be a dual Banach space, let \({\mathcal {X}}_0,\,{\mathcal {Y}},\,{\mathcal {Z}}\) be Banach spaces such that \({\mathcal {X}}=\overline{{\mathcal {X}}_0}^{\textrm{w}^*}\) and \(\phi : {\mathcal {X}}\rightarrow {\mathcal {Y}}^{**}\) be a \(\textrm{w}^*\)-continuous onto isometry such that \(\phi ({\mathcal {X}}_0)={\mathcal {Y}}.\) If \(u: {\mathcal {X}}_0\rightarrow {\mathcal {Z}}^*\) is a bounded linear map, then there exists a \(\textrm{w}^*\)-continuous bounded linear map \(\tilde{u}: {\mathcal {X}}\rightarrow {\mathcal {Z}}^*\) such that \(\tilde{u}|_{{\mathcal {X}}_0}=u\) and \(\Vert u\Vert =\Vert \tilde{u}\Vert .\)

Lemma 3.6

[5] Let \({\mathcal {A}}\) be a \(\,C^*\)-algebra such that \({\mathcal {A}}\subseteq {\mathcal {A}}^{**}\subseteq {\mathcal {B}}(H)\) and let \(P\in {\mathcal {A}}^\prime \) be a projection. Then there exists a \(C^*\)-algebra \({\mathcal {D}}\) and a \(*\)-isomorphism \(\alpha : {\mathcal {A}} P\rightarrow {\mathcal {D}}\) which extends to a \(*\)-isomorphism from \(\overline{{\mathcal {A}} P}^{\textrm{w}^*}\) to \({\mathcal {D}}^{**}.\)

The above lemmas are the key to pass from the case of separably acting von Neumann algebras to all \(C^*\)-algebras. Assuming that (EP) holds and following exactly the same proof procedure for Lemma 3.5, Theorem 3.6 and Corollary 3.7 in [5] we deduce that

Theorem 3.7

Under (EP), all \(C^*\)-algebras satisfy (SP).