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Given a Fell bundle
$\mathscr C\overset {q}{\to }\Xi $
over the discrete groupoid
$\Xi $
, we study the symmetry of the associated Hahn algebra
$\ell ^{\infty ,1}(\Xi \!\mid \!\mathscr C)$
in terms of the isotropy subgroups of
$\Xi $
. We prove that
$\Xi $
is symmetric (respectively hypersymmetric) if and only if all of the isotropy subgroups are symmetric (respectively hypersymmetric). We also characterise hypersymmetry using Fell bundles with constant fibres, showing that for discrete groupoids, ‘hypersymmetry’ equals ‘rigid symmetry’.
We prove a double commutant theorem for separable subalgebras of a wide class of corona C*-algebras, largely resolving a problem posed by Pedersen in 1988. Double commutant theorems originated with von Neumann, whose seminal result evolved into an entire field now called von Neumann algebra theory. Voiculescu later proved a C*-algebraic double commutant theorem for subalgebras of the Calkin algebra. We prove a similar result for subalgebras of a much more general class of so-called corona C*-algebras.
We resolve the isomorphism problem for tensor algebras of unital multivariable dynamical systems. Specifically, we show that unitary equivalence after a conjugation for multivariable dynamical systems is a complete invariant for complete isometric isomorphisms between their tensor algebras. In particular, this settles a conjecture of Davidson and Kakariadis, Inter. Math. Res. Not.2014 (2014), 1289–1311 relating to work of Arveson, Acta Math.118 (1967), 95–109 from the 1960s, and extends related work of Kakariadis and Katsoulis, J. Noncommut. Geom.8 (2014), 771–787.
We prove that a Banach algebra B that is a completion of the universal enveloping algebra of a finite-dimensional complex Lie algebra
$\mathfrak {g}$
satisfies a polynomial identity if and only if the nilpotent radical
$\mathfrak {n}$
of
$\mathfrak {g}$
is associatively nilpotent in B. Furthermore, this holds if and only if a certain polynomial growth condition is satisfied on
$\mathfrak {n}$
.
Let
$C_{\||.\||}$
be an ideal of compact operators with symmetric norm
$\||.\||$
. In this paper, we extend the van Hemmen–Ando norm inequality for arbitrary bounded operators as follows: if f is an operator monotone function on
$[0,\infty)$
and S and T are bounded operators in
$\mathbb{B}(\mathscr{H}\;\,)$
such that
${\rm{sp}}(S),{\rm{sp}}(T) \subseteq \Gamma_a=\{z\in \mathbb{C} \ | \ {\rm{re}}(z)\geq a\}$
, then
The classical Gelfand–Naimark theorems provide important insight into the structure of general and of commutative
$C^*$
-algebras. It is shown that these can be generalized to certain ordered
$^*$
-algebras. More precisely, for
$\sigma $
-bounded closed ordered
$^*$
-algebras, a faithful representation as operators is constructed. Similarly, for commutative such algebras, a faithful representation as complex-valued functions is constructed if an additional necessary regularity condition is fulfilled. These results generalize the Gelfand–Naimark representation theorems to classes of
$^*$
-algebras larger than
$C^*$
-algebras, and which especially contain
$^*$
-algebras of unbounded operators. The key to these representation theorems is a new result for Archimedean ordered vector spaces V: If V is
$\sigma $
-bounded, then the order of V is induced by the extremal positive linear functionals on V.
We show that every isometric action on a Cantor set is conjugate to an inverse limit of actions on finite sets; and that every faithful isometric action by a finitely generated amenable group is residually finite.
We study the isomorphic structure of
$(\sum {\ell }_{q})_{c_{0}}\ (1< q<\infty )$
and prove that these spaces are complementably homogeneous. We also show that for any operator T from
$(\sum {\ell }_{q})_{c_{0}}$
into
${\ell }_{q}$
, there is a subspace X of
$(\sum {\ell }_{q})_{c_{0}}$
that is isometric to
$(\sum {\ell }_{q})_{c_{0}}$
and the restriction of T on X has small norm. If T is a bounded linear operator on
$(\sum {\ell }_{q})_{c_{0}}$
which is
$(\sum {\ell }_{q})_{c_{0}}$
-strictly singular, then for any
$\epsilon>0$
, there is a subspace X of
$(\sum {\ell }_{q})_{c_{0}}$
which is isometric to
$(\sum {\ell }_{q})_{c_{0}}$
with
$\|T|_{X}\|<\epsilon $
. As an application, we show that the set of all
$(\sum {\ell }_{q})_{c_{0}}$
-strictly singular operators on
$(\sum {\ell }_{q})_{c_{0}}$
forms the unique maximal ideal of
$\mathcal {L}((\sum {\ell }_{q})_{c_{0}})$
.
Let
$\mathcal {X}$
be a Banach space over the complex field
$\mathbb {C}$
and
$\mathcal {B(X)}$
be the algebra of all bounded linear operators on
$\mathcal {X}$
. Let
$\mathcal {N}$
be a nontrivial nest on
$\mathcal {X}$
,
$\text {Alg}\mathcal {N}$
be the nest algebra associated with
$\mathcal {N}$
, and
$L\colon \text {Alg}\mathcal {N}\longrightarrow \mathcal {B(X)}$
be a linear mapping. Suppose that
$p_n(x_1,x_2,\ldots ,x_n)$
is an
$(n-1)\,$
th commutator defined by n indeterminates
$x_1, x_2, \ldots , x_n$
. It is shown that L satisfies the rule
for all
$A_1, A_2, \ldots , A_n\in \text {Alg}\mathcal {N}$
if and only if there exist a linear derivation
$D\colon \text {Alg}\mathcal {N}\longrightarrow \mathcal {B(X)}$
and a linear mapping
$H\colon \text {Alg}\mathcal {N}\longrightarrow \mathbb {C}I$
vanishing on each
$(n-1)\,$
th commutator
$p_n(A_1,A_2,\ldots , A_n)$
for all
$A_1, A_2, \ldots , A_n\in \text {Alg}\mathcal {N}$
such that
$L(A)=D(A)+H(A)$
for all
$A\in \text {Alg}\mathcal {N}$
. We also propose some related topics for future research.
We show that every Lorentz sequence space
$d(\textbf {w},p)$
admits a 1-complemented subspace Y distinct from
$\ell _p$
and containing no isomorph of
$d(\textbf {w},p)$
. In the general case, this is only the second nontrivial complemented subspace in
$d(\textbf {w},p)$
yet known. We also give an explicit representation of Y in the special case
$\textbf {w}=(n^{-\theta })_{n=1}^\infty $
(
$0<\theta <1$
) as the
$\ell _p$
-sum of finite-dimensional copies of
$d(\textbf {w},p)$
. As an application, we find a sixth distinct element in the lattice of closed ideals of
$\mathcal {L}(d(\textbf {w},p))$
, of which only five were previously known in the general case.
We establish several new characterizations of amenable
$W^*$
- and
$C^*$
-dynamical systems over arbitrary locally compact groups. In the
$W^*$
-setting we show that amenability is equivalent to (1) a Reiter property and (2) the existence of a certain net of completely positive Herz–Schur multipliers of
$(M,G,\alpha )$
converging point weak* to the identity of
$G\bar {\ltimes }M$
. In the
$C^*$
-setting, we prove that amenability of
$(A,G,\alpha )$
is equivalent to an analogous Herz–Schur multiplier approximation of the identity of the reduced crossed product
$G\ltimes A$
, as well as a particular case of the positive weak approximation property of Bédos and Conti [On discrete twisted
$C^*$
-dynamical systems, Hilbert
$C^*$
-modules and regularity. Münster J. Math.5 (2012), 183–208] (generalized to the locally compact setting). When
$Z(A^{**})=Z(A)^{**}$
, it follows that amenability is equivalent to the 1-positive approximation property of Exel and Ng [Approximation property of
$C^*$
-algebraic bundles. Math. Proc. Cambridge Philos. Soc.132(3) (2002), 509–522]. In particular, when
$A=C_0(X)$
is commutative, amenability of
$(C_0(X),G,\alpha )$
coincides with topological amenability of the G-space
$(G,X)$
.
We formulate general conditions which imply that ${\mathcal L}(X,Y)$, the space of operators from a Banach space X to a Banach space Y, has $2^{{\mathfrak {c}}}$ closed ideals, where ${\mathfrak {c}}$ is the cardinality of the continuum. These results are applied to classical sequence spaces and Tsirelson-type spaces. In particular, we prove that the cardinality of the set ofclosed ideals in ${\mathcal L}\left (\ell _p\oplus \ell _q\right )$ is exactly $2^{{\mathfrak {c}}}$ for all $1<p<q<\infty $.
Since their inception in the 1930s by von Neumann, operator algebras have been used to shed light on many mathematical theories. Classification results for self-adjoint and non-self-adjoint operator algebras manifest this approach, but a clear connection between the two has been sought since their emergence in the late 1960s. We connect these seemingly separate types of results by uncovering a hierarchy of classification for non-self-adjoint operator algebras and $C^{*}$-algebras with additional $C^{*}$-algebraic structure. Our approach naturally applies to algebras arising from $C^{*}$-correspondences to resolve self-adjoint and non-self-adjoint isomorphism problems in the literature. We apply our strategy to completely elucidate this newly found hierarchy for operator algebras arising from directed graphs.
Non-amenability of ${\mathcal {B}}(E)$ has been surprisingly difficult to prove for the classical Banach spaces, but is now known for E = ℓp and E = Lp for all 1 ⩽ p < ∞. However, the arguments are rather indirect: the proof for L1 goes via non-amenability of $\ell ^\infty ({\mathcal {K}}(\ell _1))$ and a transference principle developed by Daws and Runde (Studia Math., 2010).
In this note, we provide a short proof that ${\mathcal {B}}(L_1)$ and some of its subalgebras are non-amenable, which completely bypasses all of this machinery. Our approach is based on classical properties of the ideal of representable operators on L1, and shows that ${\mathcal {B}}(L_1)$ is not even approximately amenable.
In this paper, we present a version of Fraïssé theory for categories of metric structures. Using this version, we show that every UHF algebra can be recognized as a Fraïssé limit of a class of C*-algebras of matrix-valued continuous functions on cubes with distinguished traces. We also give an alternative proof of the fact that the Jiang–Su algebra is the unique simple monotracial C*-algebra among all the inductive limits of prime dimension drop algebras.
Let ${\mathcal{A}}$ be a semisimple Banach algebra with minimal left ideals and $\text{soc}({\mathcal{A}})$ be the socle of ${\mathcal{A}}$. We prove that if $\text{soc}({\mathcal{A}})$ is an essential ideal of ${\mathcal{A}}$, then every 2-local derivation on ${\mathcal{A}}$ is a derivation. As applications of this result, we can easily show that every 2-local derivation on some algebras, such as semisimple modular annihilator Banach algebras, strongly double triangle subspace lattice algebras and ${\mathcal{J}}$-subspace lattice algebras, is a derivation.
We show that Ringrose's diagonal ideals are primitive ideals in a nest algebra (subject to the continuum hypothesis). This answers an old question of Lance and provides for the first time concrete descriptions of enough primitive ideals to obtain the Jacobson radical as their intersection. Separately, we provide a standard form for all left ideals of a nest algebra, which leads to insights into the maximal left ideals. In the case of atomic nest algebras, we show how primitive ideals can be categorized by their behaviour on the diagonal and provide concrete examples of all types.
We provide a detailed study of actions of the integers on compact quantum metric spaces, which includes general criteria ensuring that the associated crossed product algebra is again a compact quantum metric space in a natural way. Moreover, we provide a flexible set of assumptions ensuring that a continuous family of $\ast$-automorphisms of a compact quantum metric space yields a field of crossed product algebras which varies continuously in Rieffel’s quantum Gromov–Hausdorff distance. Finally, we show how our results apply to continuous families of Lip-isometric actions on compact quantum metric spaces and to families of diffeomorphisms of compact Riemannian manifolds which vary continuously in the Whitney $C^{1}$-topology.
In this paper we show that every non-cycle finite transitive directed graph has a Cuntz–Krieger family whose WOT-closed algebra is $B(\mathcal {H})$. This is accomplished through a new construction that reduces this problem to in-degree 2-regular graphs, which is then treated by applying the periodic Road Colouring Theorem of Béal and Perrin. As a consequence we show that finite disjoint unions of finite transitive directed graphs are exactly those finite graphs which admit self-adjoint free semigroupoid algebras.
The notion of quantized characters was introduced in our previous paper as a natural quantization of characters in the context of asymptotic representation theory for quantum groups. As in the case of ordinary groups, the representation associated with any extreme quantized character generates a von Neumann factor. From the viewpoint of operator algebras (and measurable dynamical systems), it is natural to ask what is the Murray–von Neumann–Connes type of the resulting factor. In this paper, we give a complete solution to this question when the inductive system is of quantum unitary groups $U_{q}(N)$.