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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
$U_{q}(N)$
.
We initiate a study of structural properties of the quotient algebra
${\mathcal{K}}(X)/{\mathcal{A}}(X)$
of the compact-by-approximable operators on Banach spaces
$X$
failing the approximation property. Our main results and examples include the following: (i) there is a linear isomorphic embedding from
$c_{0}$
into
${\mathcal{K}}(Z)/{\mathcal{A}}(Z)$
, where
$Z$
belongs to the class of Banach spaces constructed by Willis that have the metric compact approximation property but fail the approximation property, (ii) there is a linear isomorphic embedding from a nonseparable space
$c_{0}(\unicode[STIX]{x1D6E4})$
into
${\mathcal{K}}(Z_{FJ})/{\mathcal{A}}(Z_{FJ})$
, where
$Z_{FJ}$
is a universal compact factorisation space arising from the work of Johnson and Figiel.
We study linear mappings which preserve vectors at a specific angle. We introduce the concept of
$(\unicode[STIX]{x1D700},c)$
-angle preserving mappings and define
$\widehat{\unicode[STIX]{x1D700}}\,(T,c)$
as the ‘smallest’ number
$\unicode[STIX]{x1D700}$
for which
$T$
is an
$(\unicode[STIX]{x1D700},c)$
-angle preserving mapping. We derive an exact formula for
$\widehat{\unicode[STIX]{x1D700}}\,(T,c)$
in terms of the norm
$\Vert T\Vert$
and the minimum modulus
$[T]$
of
$T$
. Finally, we characterise approximately angle preserving mappings.
Let
$n$
be a positive integer. A
$C^{\ast }$
-algebra is said to be
$n$
-subhomogeneous if all its irreducible representations have dimension at most
$n$
. We give various approximation properties characterising
$n$
-subhomogeneous
$C^{\ast }$
-algebras.
Let 𝔻n be the open unit polydisc in ℂn,
$n \ges 1$
, and let H2(𝔻n) be the Hardy space over 𝔻n. For
$n\ges 3$
, we show that if θ ∈ H∞(𝔻n) is an inner function, then the n-tuple of commuting operators
$(C_{z_1}, \ldots , C_{z_n})$
on the Beurling type quotient module
${\cal Q}_{\theta }$
is not essentially normal, where
Rudin's quotient modules of H2(𝔻2) are also shown to be not essentially normal. We prove several results concerning boundary representations of C*-algebras corresponding to different classes of quotient modules including doubly commuting quotient modules and homogeneous quotient modules.
Motivated by the definition of a semigroup compactication of a locally compact group and a large collection of examples, we introduce the notion of an (operator) homogeneous left dual Banach algebra (HLDBA) over a (completely contractive) Banach algebra
$A$
. We prove a Gelfand-type representation theorem showing that every HLDBA over A has a concrete realization as an (operator) homogeneous left Arens product algebra: the dual of a subspace of
$A^{\ast }$
with a compatible (matrix) norm and a type of left Arens product
$\Box$
. Examples include all left Arens product algebras over
$A$
, but also, when
$A$
is the group algebra of a locally compact group, the dual of its Fourier algebra. Beginning with any (completely) contractive (operator)
$A$
-module action
$Q$
on a space
$X$
, we introduce the (operator) Fourier space
$({\mathcal{F}}_{Q}(A^{\ast }),\Vert \cdot \Vert _{Q})$
and prove that
$({\mathcal{F}}_{Q}(A^{\ast })^{\ast },\Box )$
is the unique (operator) HLDBA over
$A$
for which there is a weak
$^{\ast }$
-continuous completely isometric representation as completely bounded operators on
$X^{\ast }$
extending the dual module representation. Applying our theory to several examples of (completely contractive) Banach algebras
$A$
and module operations, we provide new characterizations of familiar HLDBAs over A and we recover, and often extend, some (completely) isometric representation theorems concerning these HLDBAs.
We consider a linear operator pencil with complex parameter mapping one Hilbert space onto another. It is known that the resolvent is analytic in an open annular region of the complex plane centred at the origin if and only if the coefficients of the Laurent series satisfy a doubly-infinite set of left and right fundamental equations and are suitably bounded. If the resolvent has an isolated singularity at the origin we propose a recursive orthogonal decomposition of the domain and range spaces that enables us to construct the key nonorthogonal projections that separate the singular and regular components of the resolvent and subsequently allows us to find a formula for the basic solution to the fundamental equations. We show that each Laurent series coefficient in the singular part of the resolvent can be approximated by a weakly convergent sequence of finite-dimensional matrix operators and we show how our analysis can be extended to find a global expression for the resolvent of a linear pencil in the case where the resolvent has only a finite number of isolated singularities.
We generalize the notion of summable Szlenk index from a Banach space to an arbitrary weak*-compact set. We prove that a weak*-compact set has summable Szlenk index if and only if its weak*-closed, absolutely convex hull does. As a consequence, we offer a new, short proof of a result from Draga and Kochanek [J. Funct. Anal. 271 (2016), 642–671] regarding the behavior of summability of the Szlenk index under c0 direct sums. We also use this result to prove that the injective tensor product of two Banach spaces has summable Szlenk index if both spaces do, which answers a question from Draga and Kochanek [Proc. Amer. Math. Soc. 145 (2017), 1685–1698]. As a final consequence of this result, we prove that a separable Banach space has summable Szlenk index if and only if it embeds into a Banach space with an asymptotic c0 finite dimensional decomposition, which generalizes a result from Odell et al. [Q. J. Math. 59, (2008), 85–122]. We also introduce an ideal norm
$\mathfrak{s}$
on the class
$\mathfrak{S}$
of operators with summable Szlenk index and prove that
$(\mathfrak{S}, \mathfrak{s})$
is a Banach ideal. For 1 ⩽ p ⩽ ∞, we prove precise results regarding the summability of the Szlenk index of an ℓp direct sum of a collection of operators.
Given a Banach operator ideal
$\mathcal A$
, we investigate the approximation property related to the ideal of
$\mathcal A$
-compact operators,
$\mathcal K_{\mathcal A}$
-AP. We prove that a Banach space X has the
$\mathcal K_{\mathcal A}$
-AP if and only if there exists a λ ≥ 1 such that for every Banach space Y and every R ∈
$\mathcal K_{\mathcal A}$
(Y, X),
For a surjective, maximal and right-accessible Banach operator ideal
$\mathcal A$
, we prove that a Banach space X has the
$\mathcal K_{(\mathcal A^{{\rm adj}})^{{\rm dual}}}$
-AP if the dual space of X has the
$\mathcal K_{\mathcal A}$
-AP.
Triangular algebras, and maximal triangular algebras in particular, have been objects of interest for over 50 years. Rich families of examples have been studied in the context of many w*- and C*-algebras, but there remains a dearth of concrete examples in
$B({\cal H})$
. In previous work, we described a family of maximal triangular algebras of finite multiplicity. Here, we investigate a related family of maximal triangular algebras with infinite multiplicity, and unearth a new asymptotic structure exhibited by these algebras.
We add to the list of Banach spaces X for which it is known that the space of bounded linear operators on X has a unique maximal ideal. In particular, the result holds if X is a subsymmetric direct sum of ℓp or of the Schlumprecht space S. We also show that two recently identified ideals in L(Jp), where Jp is the pth James space, each contains a unique maximal ideal.
The main purpose of this paper is to investigate some natural problems regarding the order structure of representable functionals on *-algebras. We describe the extreme points of order intervals, and give a non-trivial sufficient condition to decide whether or not the infimum of two representable functionals exists. To this aim, we offer a suitable approach to the Lebesgue decomposition theory, which is in complete analogy with the one developed by Ando in the context of positive operators. This tight analogy allows to invoke Ando's results to characterize uniqueness of the decomposition, and solve the infimum problem over certain operator algebras.
We prove a necessary and sufficient condition for embeddability of an operator system into
${\mathcal{O}}_{2}$
. Using Kirchberg’s theorems on a tensor product of
${\mathcal{O}}_{2}$
and
${\mathcal{O}}_{\infty }$
, we establish results on their operator system counterparts
${\mathcal{S}}_{2}$
and
${\mathcal{S}}_{\infty }$
. Applications of the results, including some examples describing
$C^{\ast }$
-envelopes of operator systems, are also discussed.
We consider the spectral behavior and noncommutative geometry of commutators
$[P,f]$
, where
$P$
is an operator of order 0 with geometric origin and
$f$
a multiplication operator by a function. When
$f$
is Hölder continuous, the spectral asymptotics is governed by singularities. We study precise spectral asymptotics through the computation of Dixmier traces; such computations have only been considered in less singular settings. Even though a Weyl law fails for these operators, and no pseudodifferential calculus is available, variations of Connes’ residue trace theorem and related integral formulas continue to hold. On the circle, a large class of nonmeasurable Hankel operators is obtained from Hölder continuous functions
$f$
, displaying a wide range of nonclassical spectral asymptotics beyond the Weyl law. The results extend from Riemannian manifolds to contact manifolds and noncommutative tori.
The Birkhoff orthogonality has been recently intensively studied in connection with the geometry of Banach spaces and operator theory. The main aim of this paper is to characterize the Birkhoff orthogonality in
${\mathcal{L}}(X;Y)$
under the assumption that
${\mathcal{K}}(X;Y)$
is an
$M$
-ideal in
${\mathcal{L}}(X;Y)$
. Moreover, we survey the known results, as well as giving some new and more general ones. Furthermore, we characterize an approximate Birkhoff orthogonality in
${\mathcal{K}}(X;Y)$
.
Let be a single vertex k-graph and let be the von Neumann algebra induced from the Gelfand–Naimark–Segal (GNS) representation of a distinguished state ω of its k-graph C*-algebra . In this paper we prove the factoriality of , and furthermore determine its type when either has the little pullback property, or the intrinsic group of has rank 0. The key step to achieving this is to show that the fixed-point algebra of the modular action corresponding to ω has a unique tracial state.
We establish a spectral characterization theorem for the operators on complex Hilbert spaces of arbitrary dimensions that attain their norm on every closed subspace. The class of these operators is not closed under addition. Nevertheless, we prove that the intersection of these operators with the positive operators forms a proper cone in the real Banach space of hermitian operators.
A locally compact group G is compact if and only if its convolution algebras contain non-zero (weakly) completely continuous elements. Dually, G is discrete if its function algebras contain non-zero completely continuous elements. We prove non-commutative versions of these results in the case of locally compact quantum groups.
We prove that an operator system is (min, ess)-nuclear if its
$C^{\ast }$
-envelope is nuclear. This allows us to deduce that an operator system associated to a generating set of a countable discrete group by Farenick et al. [‘Operator systems from discrete groups’, Comm. Math. Phys.329(1) (2014), 207–238] is (min, ess)-nuclear if and only if the group is amenable. We also make a detailed comparison between ess and other operator system tensor products and show that an operator system associated to a minimal generating set of a finitely generated discrete group (respectively, a finite graph) is (min, max)-nuclear if and only if the group is of order less than or equal to three (respectively, every component of the graph is complete).