Amenable Operator Research Articles

Symmetric operator amenability of operator algebras

Using the notion of a symmetric virtual diagonal for a Banach algebra, we prove that a Banach algebra is symmetrically amenable if its second dual is symmetrically amenable. We introduce symmetric operator amenability in the category of completely contractive Banach algebras as an operator algebra analogue of symmetric amenability of Banach algebras. We give some equivalent formulations of symmetric operator amenability of completely contractive Banach algebras and investigate some hereditary properties of symmetric operator amenable algebras. We show that amenability of locally compact groups is equivalent to symmetric operator amenability of its Fourier algebra. Finally, we discuss about Jordan derivation on symmetrically operator amenable algebras.

On operator amenability of Fourier-Stieltjes algebras

We consider the Fourier-Stieltjes algebra B(G) of a locally compact group G. We show that operator amenability of B(G) implies that a certain semitopological compactification of G admits only finitely many idempotents. In the case that G is connected, we show that operator amenability of B(G) entails that G is compact.

Compact ideals and rigidity of representations for amenable operator algebras

We examine rigidity phenomena for representations of amenable operator algebras which have an ideal of compact operators. We establish that a generalized version of Kadison’s conjecture on completely bounded homomorphisms holds for the algebra if the asso

Operator Connes-amenability of completely bounded multiplier Banach algebras

For a completely contractive Banach algebra $B$, we find conditions under which the completely bounded multiplier algebra $\mathcal{M}_{cb}(B)$ is a dual Banach algebra and the operator amenability of $B$ is equivalent to the operator Connes-amenability of $\mathcal{M}_{cb}(B)$. We also show that, in this case, these are equivalent to the existence of a normal virtual operator diagonal.

Residual finite dimensionality and representations of amenable operator algebras

We consider a version of a famous open problem formulated by Kadison, asking whether bounded representations of operator algebras are automatically completely bounded. We investigate this question in the context of amenable operator algebras, and we provide an affirmative answer for representations whose range is residually finite-dimensional. Furthermore, we show that weak- ⁎ closed, amenable, residually finite-dimensional operator algebras are similar to C ⁎ -algebras, and in particular have the property that all their bounded representations are completely bounded. We prove our results for operator algebras having the so-called total reduction property, which is known to be weaker than amenability.

$p$-Fourier algebras on compact groups

Let G be a compact group. For 1\leq p\leq\infty we introduce a class of Banach function algebras \mathcal A^p(G) on G which are the Fourier algebras in the case p=1 , and for p=2 are certain algebras discovered by Forrest, Samei and Spronk. In the case p\neq 2 we find that \mathcal A^p(G)\cong\mathcal A^p(H) if and only if G and H are isomorphic compact groups. These algebras admit natural operator space structures, and also weighted versions, which we call p -Beurling–Fourier algebras. We study various amenability and operator amenability properties, Arens regularity and representability as operator algebras. For a connected Lie G and p > 1 , our techniques of estimation of when certain p -Beurling–Fourier algebras are operator algebras rely more on the fine structure of G , than in the case p=1 . We also study restrictions to subgroups. In the case that G= SU(2), restrict to a torus and obtain some exotic algebras of Laurent series. We study amenability properties of these new algebras, as well.

Module operator virtual diagonals on the Fourier algebra of an inverse semigroup

For an amenable inverse semigroup S with the set of idempotents E and a minimal idempotent, we explicitly construct a contractive and positive module operator virtual diagonal on the Fourier algebra A(S), as a completely contractive Banach algebra and operator module over $$\ell ^1(E)$$ . This generalizes a well known result of Zhong-Jin Ruan on operator amenability of the Fourier algebra of a (discrete) group Ruan (Am J Math 117:1449–1474, 1995).

Abelian, amenable operator algebras are similar to C∗-algebras

Suppose that H is a complex Hilbert space and that B(H) denotes the bounded linear operators on H. We show that every abelian, amenable operator algebra is similar to a C*-algebra. We do this by showing that if A is an abelian subalgebra of B(H) with the property that given any bounded representation $\varrho: A \to B(H_\varrho)$ of A on a Hilbert space $H_\varrho$, every invariant subspace of $\varrho(A)$ is topologically complemented by another invariant subspace of $\varrho(A)$, then A is similar to an abelian $C^*$-algebra.

AN ALGEBRA WHOSE SUBALGEBRAS ARE CHARACTERIZED BY DENSITY

AbstractWe refine a construction of Choi, Farah, and Ozawa to build a nonseparable amenable operator algebra ${\rm {\cal A}}$ ⊆ ℓ∞ (M2) whose nonseparable subalgebras, including ${\rm {\cal A}}$, are not isomorphic to a C*-algebra. This is done using a Luzin gap and a uniformly bounded group representation.Next, we study additional properties of ${\rm {\cal A}}$ and of its separable subalgebras, related to the Kadison Kastler metric.

A Hilbert space approach to approximate diagonals for locally compact quantum groups

For a locally compact quantum group $\mathbb{G}$, the quantum group algebra $L^1(\mathbb{G})$ is operator amenable if and only if it has an operator bounded approximate diagonal. It is known that if $L^1(\mathbb{G})$ is operator biflat and has a bounded approximate identity then it is operator amenable. In this paper, we consider nets in $L^2(\mathbb{G})$ which suffice to show these two conditions and combine them to make an approximate diagonal of the form $\omega_{{W'}^*\xi\otimes\eta}$ where $W$ is the multiplicative unitary and $\xi\otimes\eta$ are simple tensors in $L^2(\mathbb{G})\otimes L^2(\mathbb{G})$. Indeed, if $L^1(\mathbb{G})$ and $L^1(\hat{\mathbb{G}})$ both have a bounded approximate identity and either of the corresponding nets in $L^2(\mathbb{G})$ satisfies a condition generalizing quasicentrality then this construction generates an operator bounded approximate diagonal. In the classical group case, this provides a new method for constructing approximate diagonals emphasizing the relation between the operator amenability of the group algebra $L^1(G)$ and the Fourier algebra $A(G)$.

Module operator amenability of the Fourier algebra of an inverse semigroup

We define module operator amenability for completely contractive Banach algebras which are Banach modules over another Banach algebra with compatible actions and study module operator amenability of the Fourier algebra $$A(S)$$ of an inverse semigroup $$S$$ with the set of idempotents $$E$$ , as a module over the semigroup algebra $$\ell ^{1}(E)$$ . We show that when $$E$$ has a minimum element, module operator amenability of $$A(S)$$ is equivalent to amenability of $$S$$ .

A NONSEPARABLE AMENABLE OPERATOR ALGEBRA WHICH IS NOT ISOMORPHIC TO A -ALGEBRA

AbstractIt has been a long-standing question whether every amenable operator algebra is isomorphic to a (necessarily nuclear) $\mathrm{C}^*$-algebra. In this note, we give a nonseparable counterexample. Finding out whether a separable counterexample exists remains an open problem. We also initiate a general study of unitarizability of representations of amenable groups in $\mathrm{C}^*$-algebras and show that our method cannot produce a separable counterexample.

Amenable Operators of the Form Normal Plus Compact

An open question, raised independently by several authors, asks if a closed amenable subalgebra of \({\mathfrak{B}(\mathfrak{H})}\) must be similar to an C *-algebra. Recently, Choi, Farah and Ozawa have found a counter-example to this question, but their example is neither separable nor commutative, which leaves the question open for singly-generated algebras. In this paper we continue this line of investigation for special singly-generated algebras. It is shown that if an amenable operator T = N + K, where N is a normal operator, K is a compact operator and σ e (N) has only finite accumulation points, then T is similar to a normal operator; if an amenable operator T = N + K, where N is a normal operator, \({K\in\mathcal{C}_p}\) for some p > 1 and \({\sigma(T)\cup\sigma(N)}\) is included in a smooth Jordan curve, then T is similar to a normal operator; if an amenable operator T = N + Q, where N is a normal operator, Q is a polynomial compact operator and NQ = QN, then T is similar to a normal operator; if there exists p, 1 < p < ∞, such that an amenable operator T satisfies one of the following conditions, then T is similar to a normal operator: (i) \({T-T^*\in\mathcal{C}_p}\); (ii) \({I-TT^*\in\mathcal{C}_p}\) ; (iii)\({I-T^*T\in\mathcal{C}_p}\).

Semisimplicity of reflexive amenable operator algebras

We prove that amenable and reflexive operator algebras are semisimple and are finite direct sums of simple Banach algebras of operators.

Operator biflatness of the L 1-algebras of compact quantum groups

Abstract We prove that the L 1-algebra of any non-Kac type compact quantum group is not operator biflat. Since operator amenability implies operator biflatness, this result shows that any co-amenable, non-Kac type compact quantum group gives a counterexample to the conjecture that L 1(𝔾) is operator amenable if and only if 𝔾 is amenable and co-amenable for any locally compact quantum group 𝔾. The result also implies that the L 1-algebra of a locally compact quantum group is operator biprojective if and only if 𝔾 is compact and of Kac type.

Topological Wiener–Wintner theorems for amenable operator semigroups

AbstractInspired by topological Wiener–Wintner theorems we study the mean ergodicity of amenable semigroups of Markov operators on $C(K)$ and show the connection to the convergence of strong and weak ergodic nets. The results are then used to characterize mean ergodicity of Koopman semigroups corresponding to skew product actions on compact group extensions.

Invariant and hyperinvariant subspaces for amenable operators

There has been a long-standing conjecture in Banach algebras that every amenable operator is similar to a normal operator. In this paper, we study the structure of amenable operators on Hilbert spaces. At first, we show that the conjecture is equivalent to the statement that every non-scalar amenable operator has a non-trivial hyperinvariant subspace. It is also equiv- alent to the statement that every amenable operator is similar to a reducible operator and has a non-trivial invariant subspace; and then, we give two de- compositions for amenable operators, supporting the conjecture.

Uniform families of ergodic operator nets

We study mean ergodicity in amenable operator semigroups and establish the connection to the convergence of strong and weak ergodic nets. We then use these results in order to show the convergence of uniform families of ergodic nets that appear in topological Wiener–Wintner theorems.

Beurling–Fourier algebras, operator amenability and Arens regularity

We introduce the class of Beurling–Fourier algebras on locally compact groups and show that they are non-commutative analogs of classical Beurling algebras. We obtain various results with regard to the operator amenability, operator weak amenability and Arens regularity of Beurling–Fourier algebras on compact groups and show that they behave very similarly to the classical Beurling algebras of discrete groups. We then apply our results to study explicitly the Beurling–Fourier algebras on SU ( 2 ) , the 2 × 2 special unitary group. We demonstrate that how Beurling–Fourier algebras are closely connected to the amenability of the Fourier algebra of SU ( 2 ) . Another major consequence of our results is that our investigation allows us to construct families of unital infinite-dimensional closed Arens regular subalgebras of the Fourier algebra of certain products of SU ( 2 ) .

Near inclusions of amenable operator algebras

We prove that if an amenable operator algebra is nearly contained in a complemented dual operator algebra, then it can be embedded inside this dual operator algebra via a similarity. The proof relies on a Johnson Theorem on approximately multiplicative maps.