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C*-Algebras and Their Automorphism Groups
... The theorem states that, under suitable assumptions on a Γ-C * -algebra inclusion B ⊂ (A, α), after tensoring with the Cuntz algebra O 2 (equipped with the trivial action), all intermediate C * -algebras of the inclusion B ⊂ A ⋊ r,α Γ split into a twisted crossed product in a canonical way. As many deep structural results are available for the (twisted) crossed products (see the books [46], [10], [63], [3] for instance) and the associated (unique) cocycle actions are given on the purely algebraic level, the result is helpful to understand the inclusions of the above form. The proof is given in Section 3. We extend Theorem A to twisted and/or non-unital case in Section 4 (see Theorem 4.4). ...
... (See Section 8.1 in [46] for instance.) When χ = 1, A 1 is nothing but the fixed point algebra A K . ...
... Set H := τ ∈Υ ker(τ ). Since A = span{A τ : τ ∈ Υ} (see e.g., Section 8.1 in [46]), one has A H = A. By (the injectivity of) the Galois correspondence, one has H = {1}. Hence Υ = K. ...
We investigate the C*-algebra inclusions $B \subset A \rtimes_{\rm r} \Gamma$ arising from inclusions $B \subset A$ of $\Gamma$-C*-algebras. The main result shows that, when $B \subset A$ is C*-irreducible in the sense of R{\o}rdam, and is centrally $\Gamma$-free in the sense of the author, then after tensoring with the Cuntz algebra $\mathcal{O}_2$, all intermediate C*-algebras $B \subset C\subset A \rtimes_{\rm r} \Gamma$ enjoy a natural crossed product splitting \[\mathcal{O}_2\otimes C=(\mathcal{O}_2 \otimes D) \rtimes_{{\rm r}, \gamma, \mathfrak{w}} \Lambda\] for $D:= C \cap A$, some $\Lambda<\Gamma$, and a subsystem $(\gamma, \mathfrak{w})$ of a unitary perturbed cocycle action $\Lambda \curvearrowright \mathcal{O}_2\otimes A$. As an application, we give a new Galois's type theorem for the Bisch--Haagerup type inclusions \[A^K \subset A\rtimes_{\rm r} \Gamma\] for actions of compact-by-discrete groups $K \rtimes \Gamma$ on simple C*-algebras. Due to a K-theoretical obstruction, the operation $\mathcal{O}_2\otimes -$ is necessary to obtain the clean splitting. Also, in general 2-cocycles $\mathfrak{w}$ appearing in the splitting cannot be removed even further tensoring with any unital (cocycle) action. We show them by examples, which further show that $\mathcal{O}_2$ is a minimal possible choice. We also establish a von Neumann algebra analogue, where $\mathcal{O}_2$ is replaced by the type I factor $\mathbb{B}(\ell^2(\mathbb{N}))$.
... Assume from now on that is abelian. We follow the convention of [19,30] by using the inverse Fourier transform on , that is, lettingf ...
... In this subsection, we follow [19,Sections 2,3], which is based on [25], see also [30,Section 7.8]. Let ρ A be an action of a locally compact abelian group on a C *algebra A and let be a continuous 2-cocycle on the dualˆ . ...
... this again follows the convention of [19,30] in contrast to the common convention [16,41] ...
Consider a locally compact quantum group G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {G}$$\end{document} with a closed classical abelian subgroup Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document} equipped with a 2-cocycle Ψ:Γ^×Γ^→C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi :\hat{\Gamma }\times \hat{\Gamma }\rightarrow \mathbb {C}$$\end{document}. We study in detail the associated Rieffel deformation GΨ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {G}^{\Psi }$$\end{document} and establish a canonical correspondence between Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}-invariant convolution semigroups of states on G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {G}$$\end{document} and on GΨ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {G}^{\Psi }$$\end{document}.
... We review here some concepts and results on C * -algebras and representation theory that will be needed in the paper. For general references, see for instance [11,13,18]. ...
... Assume that for all 0 = a ∈ A and 0 = b ∈B we have (18) ab = 0, then (B2) holds on B = alg(A, U G ) and id ⋊ U : A ⋊ α G → B is an isomorphism. ...
... Thus we have that A ∼ = C(X) and B ∼ = C(Y ). Since z V commutes with u we have that C := alg{A, B} ∼ = C(Z) is commutative, and by condition (18), since A ⊂ A and B ⊂B, for all 0 = a ∈ A and b ∈ B we have ab = 0. Thus we can apply Lemma 3.12. We consider x ∈ V such that ...
The local trajectories method establishes invertibility in an algebra $\mathcal{B}= \operatorname{alg}(\mathcal{A}, U_G)$, for a unital $C^*$-algebra $\mathcal{A}$ and a unitary group action $U_g$, $g\in G$, of a discrete amenable group $G$ on $\mathcal{A}$. We introduce here a local type condition that allows to establish an isomorphism between $\mathcal{B}$ and a $C^*$-crossed product, which is fundamental for the method to work. The influence of the structure of the fixed points of the group action is analysed and a new equivalent condition is introduced that applies when the action is not topologically free. If $\mathcal{A}$ is commutative, the referred conditions are related to the subalgebra $\operatorname{alg}(U_G)$ yielding, in particular, a sufficient condition that depends only on the action. It is shown that in $\pi(\mathcal{B})$, with $\pi$ the local trajectories representation, the local type condition is verified, which allows establishing the isomorphism on the local trajectories method. It is analysed whether the family of representations of the local trajectories method is sufficient using the notions of strictly norming and exhaustive families. It is shown that if $\mathcal{A}$ is a commutative algebra or a matrix algebra of continuous functions the family is sufficient.
... A multitude of references have been devoted to this object. Let us briefly recall that for each C * -algebra A, the multiplier algebra M(A) of A can be defined as the idealizer of A in its bidual A * * (i.e., the largest C * -subalgebra of the von Neumann algebra A * * containing A as an ideal), equivalently, [4,19] and [47,Sect. 3.12]). The multiplier algebra of a C * -algebra is introduced with the aim of finding an appropriate unital extension within the smallest one obtained by adjoining a unit to A, and the largest natural one given by its second dual. ...
... As we have commented before, surjective Jordan * -homomorphisms, triple homomorphisms and continuous linear orthogonality preserving operators between JB * -algebras can be extended to similar types of maps between their multiplier algebras, however the extension is not, in general surjective. In the setting of C * -algebras G.K. Pedersen proved that every surjective * -homomorphism between σ -unital C * -algebras extends to a surjective * -homomorphism between their corresponding multiplier algebras (see [48,Theorem 10] or [47,Proposition 3.12.10]). The main goal in Sect. 4 is to prove an appropriate version of Pedersen's result for JB * -algebras, the desired conclusions are achieved in Theorem 4.4, Proposition 4.5 and Corollary 4.6. ...
... This section is devoted to establish a first application of the J-strict topology in a result guaranteeing when a surjective Jordan * -homomorphism between two JB * -algebras admits an extension to a surjective Jordan * -homomorphism between the multiplier algebras. In the setting of C * -algebras G. K. Pedersen was the first one observing that every surjective * -homomorphism between σ -unital C * -algebras extends to a surjective * -homomorphism between their corresponding multiplier algebras (see [48,Theorem 10] or [47,Proposition 3.12.10]). The hypothesis affirming that A and B are σ -unital cannot be relaxed (see [47, 3.12.11] ...
We introduce the Jordan-strict topology on the multiplier algebra of a JB $$^*$$ ∗ -algebra, a notion which was missing despite the forty years passed after the first studies on Jordan multipliers. In case that a C $$^*$$ ∗ -algebra A is regarded as a JB $$^*$$ ∗ -algebra, the J-strict topology of M ( A ) is precisely the well-studied C $$^*$$ ∗ -strict topology. We prove that every JB $$^*$$ ∗ -algebra $${\mathfrak {A}}$$ A is J-strict dense in its multiplier algebra $$M({\mathfrak {A}})$$ M ( A ) , and that latter algebra is J-strict complete. We show that continuous surjective Jordan homomorphisms, triple homomorphisms, and orthogonality preserving operators between JB $$^*$$ ∗ -algebras admit J-strict continuous extensions to the corresponding type of operators between the multiplier algebras. We characterize J-strict continuous functionals on the multiplier algebra of a JB $$^*$$ ∗ -algebra $${\mathfrak {A}}$$ A , and we establish that the dual of $$M({\mathfrak {A}})$$ M ( A ) with respect to the J-strict topology is isometrically isomorphic to $${\mathfrak {A}}^*$$ A ∗ . We also present a first application of the J-strict topology of the multiplier algebra, by showing that under the extra hypothesis that $${\mathfrak {A}}$$ A and $${\mathfrak {B}}$$ B are $$\sigma $$ σ -unital JB $$^*$$ ∗ -algebras, every surjective Jordan $$^*$$ ∗ -homomorphism (respectively, triple homomorphism or continuous orthogonality preserving operator) from $${\mathfrak {A}}$$ A onto $${\mathfrak {B}}$$ B admits an extension to a surjective J-strict continuous Jordan $$^*$$ ∗ -homomorphism (respectively, triple homomorphism or continuous orthogonality preserving operator) from $$M({\mathfrak {A}})$$ M ( A ) onto $$M({\mathfrak {B}})$$ M ( B ) .
... A notion of C * -algebra is explained in [11,12,14]. ...
... Theorem C.1. (Dauns Hofmann) [12] For each C * -algebra A there is the natural isomorphism from the center of M (A) onto the class of bounded continuous functions onǍ. ...
... Definition C.2. [12] Let A be a C * -algebra with the spectrumÂ. We choose for any t ∈Â a pure state φ t and associated representation π t : A → B (H t ). ...
For any topological space there is a sheaf cohomology. A Grothendieck topology is a generalization of the classical topology such that it also possesses a sheaf cohomology. On the other hand any noncommutative $C^*$-algebra is a generalization of a locally compact Hausdorff space. Here we define a Grothendieck topology of $C^*$-algebras which is a generalization of the topology of the spectra of commutative $C^*$-algebras. This construction yields a noncommutative generalization of the sheaf cohomology of topological spaces.
... We begin with the introduction of a new tool called the generalized polar decomposition. For every T ∈ L(H , K ) and every number α ∈ (0, 1), the generalized polar decomposition of T with respect to the parameter α is provided in Theorem 3.1 by following the techniques employed in the proofs of [23,Lemma 1.4.4 and Proposition 1.4.5]. Such a new tool is particularly useful in dealing with some inequalities related to Hilbert C * -modules. ...
... By the proof of [23,Proposition 1.4.5], {U n } ∞ n=1 converges to some U ∈ L(H ) in the norm topology such that ...
This paper deals mainly with some aspects of the adjointable operators on Hilbert \(C^*\)-modules. A new tool called the generalized polar decomposition for each adjointable operator is introduced and clarified. As an application, the general theory of the weakly complementable operators is set up in the framework of Hilbert \(C^*\)-modules. It is proved that there exists an operator equation which has a unique solution, whereas this unique solution fails to be the reduced solution. Some investigations are also carried out in the Hilbert space case. It is proved that there exist a closed subspace M of certain Hilbert space K and an operator \(T\in {\mathbb {B}}(K)\) such that T is (M, M)-weakly complementable, whereas T fails to be (M, M)-complementable. The solvability of the equation
is also dealt with in the Hilbert space case, where \(A,B\in {\mathbb {B}}(H)\) are two general positive operators, and A : B denotes their parallel sum. Among other things, it is shown that there exist certain positive operators A and B on the Hilbert space \(\ell ^2({\mathbb {N}})\oplus \ell ^2({\mathbb {N}})\) such that the above equation has no solution.
... Unless otherwise specified, all our groups are discrete and countable, their identity elements are denoted by the same symbol e, and all topological groupoids are second countable, locally compact and Hausdorff. We refer to [23,28] for more details on topological groupoids and their C * -algebras, and refer to [19,32] for C * -dynamical systems. ...
... for ξ, η ∈ C c (G, C * r (R)). The reduced crossed product C * -algebra, denoted by C * r (R) ⋊ α,r G, associated to the C * -dynamical system is defined to be the closure of C c (G, C * r (R)) under the reduced crossed norm (see [19,32]). By identifying an element a ∈ C * r (R) with the element ξ a ∈ C c (G, C * r (R)) defined by ξ a (e) = a and ξ a (g) = 0 for g ̸ = e, C * r (R) can be embedded into C * r (R) ⋊ a,r G as a unital C * -subalgebra. ...
... The space L p (L G) is defined as the closure of B(ℓ 2 (G)) respect to this norm. All of this can be done in more generality for non-discrete groups, using the Haar measure of G and defining a weight τ instead of a trace, see [Ped79]. The L p -spaces over von Neumann algebras can also be defined in more generality, see for example [PX03]. ...
In a recent work by Gonz\'alez-P\'erez, Parcet and Xia, the boundedness over non-commutative $L_p$-spaces of an analogue of the Hilbert transform was studied. This analogue is defined as a Fourier multiplier with symbol $m\colon \mathrm{PSL}_2(\mathbb{C})\to \mathbb{R}$ arising from the action by isometries of $\mathrm{PSL}_2(\mathbb{C})$ on the $3$-dimensional hyperbolic space $\mathbb{H}^3$. More concretely, $m$ is a lifting of the function on $\mathbb{H}^3$ that takes values $\pm 1$ in the two regions formed by dividing the space by a hyperbolic plane. The boundedness of $T_m$ on $L_p(\mathcal{L} \mathrm{PSL}_2(\mathbb{C}))$ for $p\neq 2$ was disproved by Parcet, de la Salle and Tablate. Nevertheless, we will show that this Fourier multiplier is bounded when restricted to the arithmetic lattices $\mathrm{PSL}_2(\mathbb{Z}[\sqrt{-n}])$, solving a question left open by Gonz\'alez-P\'erez, Parcet and Xia.
... For a thorough treatment of multipliers we refer to [12,Sec. 3.12]. ...
Let G be a compact group, let B be a unital C*-algebra, and let (A,G,\alpha) be a free C*-dynamical system, in the sense of Ellwood, with fixed point algebra B. We prove that (A,G,\alpha) can be realized as the invariants of an equivariant coaction of G on a corner of B \otimes \mathcal{K}(\mathfrak{H}) for a certain Hilbert space \mathfrak{H} that arises from the freeness of the action. This extends a result by Wassermann for free and ergodic C*-dynamical systems. As an application, we show that any faithful *-representation of B on a Hilbert space \mathfrak{H}_B gives rise to a faithful covariant representation of (A,G,\alpha) on some truncation of \mathfrak{H}_B \otimes \mathfrak{H}.
... First, the proof of Proposition 2.3 in [11] needs correction (the result itself is true). If B is unital then in the proof that (iii) implies (i) if a n ∈ B sa one may appeal to [38,Proposition 3.11.9] to see that 1 − q is open. If b n q then let A = C * ({1, b n }), a separable C * -algebra with q ∈ A * * . ...
We investigate some new classes of operator algebras which we call semi-σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}-finite subdiagonal and Riesz approximable. These constitute the most general setting to date for a noncommutative Hardy space theory based on Arveson’s subdiagonal algebras. We develop this theory and study the properties of these new classes.
... It is well known (see [20,Theorem 6.2.11]) that if A is a type I C * -algebra, then A contains an essential ideal that has a continuous trace. Moreover, A has a composition series {I θ | 0 ≤ θ ≤ β}, such that I θ+1 /I θ has Hausdorff spectrum for each θ < β. ...
Let $B$ be a separable $C^*$-algebra, let $\Gamma$ be a discrete
countable group, let $\alpha \colon \Gamma \to \aut(B)$ be an action, and let
$A$ be an invariant subalgebra. We find certain freeness conditions which guarantee that any
intermediate $C^*$-algebra $A \rtimes_{\alpha,r} \Gamma \subseteq C \subseteq B
\rtimes_{\alpha,r} \Gamma$ is a crossed product of an intermediate invariant
subalgebra $A \subseteq C_0 \subseteq B$ by $\Gamma$. Those are used to generalize related results by Suzuki.
... We make casual use of operator-algebraic background, as available, say, in [10,32,38] and numerous other good sources; the references are more precise where appropriate. References for locally compact quantum groups include [5,23,29], again with more specific citations in the text below. ...
... B as desired. The proof of (2) follows from a well know result that the extreme points of the set of contractions in M 2 (C) are unitary (see [41,43]). Now if we define the affine mapping f : ...
One of the first proposals for the use of quantum computers was the simulation of quantum systems. Over the past three decades, great strides have been made in the development of algorithms for simulating closed quantum systems and the more complex open quantum systems. In this tutorial, we introduce the methods used in the simulation of single qubit Markovian open quantum systems. It combines various existing notations into a common framework that can be extended to more complex open system simulation problems. The only currently available algorithm for the digital simulation of single qubit open quantum systems is discussed in detail. A modification to the implementation of the simpler channels is made that removes the need for classical random sampling, thus making the modified algorithm a strictly quantum algorithm. The modified algorithm makes use of quantum forking to implement the simpler channels that approximate the total channel. This circumvents the need for quantum circuits with a large number of CNOT gates.Quanta 2023; 12: 131–163.
... For the sake of brevity we shall not employ too much time on introducing examples nor basic results, which seem to be widely known. We refer to the monographs [15,74,100,102,108,110] as reference sources on C * -algebras. ...
In this note we collect some significant contributions on metric invariants for complex Banach algebras and Jordan--Banach algebras established during the last fifteen years. This note is mainly expository, but it also contains complete proofs and arguments, which in many cases are new or have been simplified. We have also included several new results. The common goal in the results is to seek for ``natural'' subsets, $\mathfrak{S}_{A},$ associated with each complex Banach or Jordan--Banach algebra $A$, sets which when equipped with a certain metric, $d_{A}$, enjoys the property that each surjective isometry from $(\mathfrak{S}_{A},d_A)$ to a similar set, $(\mathfrak{S}_{B},d_B),$ associated with another Banach or Jordan--Banach algebra $B$, extends to a surjective real-linear isometry from $A$ onto $B$. In case of a positive answer to this question, the problem of discussing whether in such a case the algebras $A$ and $B$ are in fact isomorphic or Jordan isomorphic is the subsequent question. The main results presented here will cover the cases in which the sets $(\mathfrak{S}_{A},d_A)$ and $(\mathfrak{S}_{B},d_B)$ are in one of the following situations: \smallskip $(\checkmark)$ Subsets of the set of invertible elements in a unital complex Banach algebra or in a unital complex Jordan--Banach algebra with the metric induced by the norm. Specially in the cases of unital C$^*$- and JB$^*$-algebras.\smallskip $(\checkmark)$ The sets of positive invertible elements in unital C$^*$- or JB$^*$-algebras with respect to the metric induced by the norm and with respect to the Thompson's metric. \smallskip $(\checkmark)$ Subsets of the set of unitary elements in unital C$^*$- and JB$^*$-algebras.
... Following [Pede,Definition 3.2.3], let us recall that two linear functionals ω 1 and ω 2 over A are said orthogonal if ...
This work is concerned with the notion of eigenstates for C*-algebras. After reviewing some basic and structural results, we explore the possibility of reinterpreting certain typical concepts of quantum mechanics (e.g., dynamical equilibrium states, ground states, gapped states, Fermi surfaces) in terms of (algebraic) eigenstates.
... Note first that we must have λ(η + η ′ ) = λ(η)λ(η ′ ). Moreover, λ must be a universally measurable function on Ξ [82,Prop. 7.4.5]. Since the only measurable characters on R 2s+n are exponentials, we conclude that λ(η) = exp(iξ·η). ...
We study continuous variable systems, in which quantum and classical degrees of freedom are combined and treated on the same footing. Thus all systems, including the inputs or outputs to a channel, may be quantum-classical hybrids. This allows a unified treatment of a large variety of quantum operations involving measurements or dependence on classical parameters. The basic variables are given by canonical operators with scalar commutators. Some variables may commute with all others and hence generate a classical subsystem. We systematically study the class of "quasifree" operations, which are characterized equivalently either by an intertwining condition for phase-space translations or by the requirement that, in the Heisenberg picture, Weyl operators are mapped to multiples of Weyl operators. This includes the well-known Gaussian operations, evolutions with quadratic Hamiltonians, and "linear Bosonic channels", but allows for much more general kinds of noise. For example, all states are quasifree. We sketch the analysis of quasifree preparation, measurement, repeated observation, cloning, teleportation, dense coding, the setup for the classical limit, and some aspects of irreversible dynamics, together with the precise salient tradeoffs of uncertainty, error, and disturbance. Although the spaces of observables and states are infinite dimensional for every non-trivial system that we consider, we treat the technicalities related to this in a uniform and conclusive way, providing a calculus that is both easy to use and fully rigorous.
... and ψ are properly asymptotically unitarily equivalent when there is a continuous path (ut) t≥0 of unitaries in I † with utφ(a)u * t → ψ(a) for all a ∈ A. 112 K 1 -injectivity allows one to know that q I (u 0 ) is in the path component of the identitywhenever [q I (u 0 )] 1 = 0.113 Indeed, since φ is absorbing, φ is unitarily equivalent to φ ⊕ φ modulo I by Corollary 5.11, and if u is a unitary implementing this equivalence and s 1 and s 2 are the Cuntz isometries defining the direct sum, then us 1 and us 2 are isometries in Q(I) ∩ q I (φ(A)) ′ with orthogonal range projections.114 Dadarlat and Eilers' strategy has its origins in the large body of work on automorphisms and derivations on operator algebras undertaken in the 1960s ([148,246,149,211]; see also[219, Sections 8.6 and 8.7]). ...
We classify the unital embeddings of a unital separable nuclear $C^*$-algebra satisfying the universal coefficient theorem into a unital simple separable nuclear $C^*$-algebra that tensorially absorbs the Jiang--Su algebra. This gives a new and essentially self-contained proof of the stably finite case of the unital classification theorem: unital simple separable nuclear $C^*$-algebras that absorb the Jiang--Su algebra tensorially and satisfy the universal coefficient theorem are classified by Elliott's invariant of $K$-theory and traces.
... It is relevant to note that the left support projection of a pure state is automatically closed. This is known to experts (see [44,Proposition 3.13.6]), but we provide additional details for the reader's convenience. ...
We study boundaries for unital operator algebras. These are sets of irreducible ∗ * -representations that completely capture the spatial norm attainment for a given subalgebra. Classically, the Choquet boundary is the minimal boundary of a function algebra and it coincides with the collection of peak points. We investigate the question of minimality for the non-commutative counterpart of the Choquet boundary and show that minimality is equivalent to what we call the Bishop property. Not every operator algebra has the Bishop property, but we exhibit classes of examples that do. Throughout our analysis, we exploit various non-commutative notions of peak points for an operator algebra. When specialized to the setting of C ∗ \mathrm {C}^* -algebras, our techniques allow us to provide a new proof of a recent characterization of those C ∗ \mathrm {C}^* -algebras admitting only finite-dimensional irreducible representations.
... Given A, B ⊂ A, de distance between A and B will be denoted by dist (A, B). See [1], [4], [11], [15] and [18] for other definitions and results about C*-algebras. ...
A polyhedron in a Banach space is a family of points $\mathcal{X}$ such that for every $x\in \mathcal{X}$, there is a closed convex set $C$ such that $a\notin C$ and $\mathcal{X}\setminus\{x\}\subset C$. In this article, we consider the notion of C*-convexity and introduce the notion of a C*-polyhedron, which is a noncommutative version of the notion of polyhedrons. We investigate the largest size of a C*-polyhedron in some classical C*-algebras.
... (Here, we identify ϕ, ψ with their normal extensions to A .) Remark 3.3. The proof of the previous lemma can also be tracked through various statements from [10], e.g. 3.6.11, ...
... Observe that α-0-KMS weights are nothing but α-invariant tracial weights. For any tracial weight τ , by the tracial condition, the dense * -subalgebra M τ ⊂ A in fact forms an algebraic ideal of A. Consequently M τ contains the Pedersen ideal Ped(A) of A. (For basic facts on Pedersen ideals, we refer the reader to Section 5.6 of [28].) ...
Associated to a family of $G$-$\ast$-endomorphisms on a $G$-C*-algebra $A$ satisfying certain minimality conditions, we give a $G$-C*-correspondence $\mathcal{E}$ over $A$ whose Cuntz--Pimsner algebra $\mathcal{O}_\mathcal{E}$ is simple. For certain quasi-free flows $\gamma$ (commuting with the $G$-action) on $\mathcal{O}_\mathcal{E}$, we further prove the simplicity of the reduced crossed product $\mathcal{O}_\mathcal{E} \rtimes_\gamma \mathbb{R}$. We then classify the KMS weights of $\gamma$. This in particular gives a sufficient condition for $\mathcal{O}_\mathcal{E}$ and $\mathcal{O}_\mathcal{E}\rtimes_\gamma \mathbb{R}$ to be stably finite (and to be stably projectionless). As the amenability of $G \curvearrowright A$ inherits to the induced actions $G \curvearrowright \mathcal{O}_\mathcal{E}, \mathcal{O}_\mathcal{E}\rtimes_\gamma \mathbb{R}$, this provides a new systematic framework to provide amenable actions on stably finite simple C*-algebras.
We introduce and study some new uniform structures for Hilbert \(C^*\)-modules over a \(C^*\)-algebra \(\mathcal {A}.\) In particular, we prove that in some cases they have the same totally bounded sets. To define one of them, we introduce a new class of \(\mathcal {A}\)-functionals: locally adjointable functionals, which have interesting properties in this context and seem to be of independent interest. A relation between these uniform structures and the theory of \(\mathcal {A}\)-compact operators is established.
This is a review of
Chirvasitu, Alexandru
Non-commutative ambits and equivariant compactifications. (English) Zbl 07828327
(Preprint available on arxiv, linked above.) We generalise techniques of Bhat–Skeide (2015) to interpolate comm. families of contractions via comm. families of contractive C₀-semigroups. Building on this, we obtain a continuous non-dilatable family, as an analogue to Parrott's discrete counter-example (1970): For any infinite dimensional Hilbert space H, there is a family of 3 commuting contractive C₀-semigroups, which admits no simultaneous unitary dilation.
This leads to some interesting residuality results within the space of contractive d-parameter contractive C₀-semigroups on a separable ∞-dim. Hilbert space H, where the space is topologised in a certain 'weak' sense:
1) the subset of unitary semigroups is dense G_δ (resp. has meagre and thus nowhere dense closure) if d ∈ {1, 2} (resp. d ≥ 3);
2) the subset of unitarily dilatable and unitarily approximable (resp. non-dilatable and non-approximable) semigroups is residual if d ∈ {1, 2} (resp. d ≥ 3).
Similar residuality results hold for the discrete setting, i.e. the space of comm. families of contractions on H, equipped with the PW-topology. We furthermore obtain the following reduction of Parrott's conjecture:
3) Let d ≥ 3. The subset of d-tuples of (commuting) contractions satisfying the von Neumann inequality is either equal to the whole space, or else is a closed meagre (thus nowhere dense) subset.
In addition we bring both the Bhat–Skeide interpolation and our 0–1 results to bear on the embedding problem. Notably, we show that the Bhat–Skeide interpolation is in a geometric sense optimal, and that generic pairs of commuting contractions can be embedded into commuting pairs of C0-semigroups, which extends Eisner's results (2009–10).
We study the gauge-invariant ideal structure of the Nica-Toeplitz algebra NT(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {N}\mathcal {T}(X)$$\end{document} of a product system (A, X) over Nn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {N}^n$$\end{document}. We obtain a clear description of X-invariant ideals in A, that is, restrictions of gauge-invariant ideals in NT(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {N\hspace{-1.111pt}T}(X)$$\end{document} to A. The main result is a classification of gauge-invariant ideals in NT(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {N\hspace{-1.111pt}T}(X)$$\end{document} for a proper product system in terms of families of ideals in A. We also apply our results to higher-rank graphs.
This is the first of two papers dedicated to the detailed determination of the reduced C*-algebra of a connected, linear, real reductive group up to Morita equivalence, and a new and very explicit proof of the Connes–Kasparov conjecture for these groups using representation theory. In this part we shall give details of the C*-algebraic Morita equivalence and then explain how the Connes–Kasparov morphism in operator K-theory may be computed using what we call the Matching Theorem, which is a purely representation-theoretic result. We shall prove our Matching Theorem in the sequel, and indeed go further by giving a simple, direct construction of the components of the tempered dual that have non-trivial K-theory using David Vogan’s approach to the classification of the tempered dual.
In this article, we introduce and study the notion of Goldie dimension for C*-algebras. We prove that a C*-algebra A has Goldie dimension n if and only if the dimension of the center of its local multiplier algebra is n. In this case, A has finite-dimensional center and its primitive spectrum is extremally disconnected. If moreover, A is extending, we show that it decomposes into a direct sum of n prime C*-algebras. In particular, every stably finite, exact C*-algebra with Goldie dimension, that has the projection property and a strictly full element, admits a full projection and a non-zero densely defined lower semi-continuous trace. Finally we show that certain C*-algebras with Goldie dimension (not necessarily simple, separable or nuclear) are classifiable by the Elliott invariant.
In this work, we present methods for constructing representations of polynomial covariance type commutation relations $$AB=BF(A)$$ A B = B F ( A ) by linear integral operators in Banach spaces $$L_p$$ L p . We derive necessary and sufficient conditions on the kernel functions for the integral operators to satisfy the covariance type commutation relation for general polynomials F , as well as for important cases, when F is arbitrary affine or quadratic polynomial, or arbitrary monomial of any degree. Using the obtained general conditions on the kernels, we construct concrete examples of representations of the covariance type commutation relations by integral operators on $$L_p$$ L p . Also, we derive useful general reordering formulas for the integral operators representing the covariance type commutation relations, in terms of the kernel functions.
For the space of \((\sigma ,\tau )\)-derivations of the group algebra \( \mathbb {C} [G] \) of a discrete countable group G, the decomposition theorem for the space of \((\sigma ,\tau )\)-derivations, generalising the corresponding theorem on ordinary derivations on group algebras, is established in an algebraic context using groupoids and characters. Several corollaries and examples describing when all \((\sigma ,\tau )\)-derivations are inner are obtained. Considered in details are cases of \((\sigma , \tau )\)-nilpotent groups and \((\sigma , \tau )\)-FC groups.
We introduce a *-module extension Banach algebras to generalized the results of Essaleh and Peralta. Precisely, let g,t and h are bounded homomorphism maps on an unital *-module extension Banach algebra, if a bounded linear map D on an unital *-module extension Banach algebra is (g,t,h)-ternary derivation at the unit element, then the next statements are hold: 1) D is (g,h)-generalized derivation; 2) D is *-(g,h)-derivation and (g,t,h)-triple (ternary) derivation, whenever D(1,0)=(0,0); 3) D is (g,t,h)-ternary derivation. In addition, we prove that a bounded linear map on *-module extension Banach algebra which is (g,h)-derivation or (g,t,h)-ternary derivation at the zero element is (g,h)-generalized derivation.
Let $\mathbb{G}$ be a locally compact quantum group, and $A,B$ von Neumann algebras with an action by $\mathbb{G}$. We refer to these as $\mathbb{G}$-dynamical W$^*$-algebras. We make a study of $\mathbb{G}$-equivariant $A$-$B$-correspondences, that is, Hilbert spaces $\mathcal{H}$ with an $A$-$B$-bimodule structure by $*$-preserving normal maps, and equipped with a unitary representation of $\mathbb{G}$ which is equivariant with respect to the above bimodule structure. Such structures are a Hilbert space version of the theory of $\mathbb{G}$-equivariant Hilbert C$^*$-bimodules. We show that there is a well-defined Fell topology on equivariant correspondences, and use this to formulate approximation properties for them. Within this formalism, we then characterize amenability of the action of a locally compact group on a von Neumann algebra, using recent results due to Bearden and Crann. We further consider natural operations on equivariant correspondences such as taking opposites, composites and crossed products, and examine the continuity of these operations with respect to the Fell topology.
Representations of polynomial covariance commutation relations by pairs of linear integral and differential operators are constructed in the space of infinitely continuously differentiable functions. Representations of polynomial covariance commutation relations by pairs consisting of a differential and linear integral operator are considered including conditions on kernels and coefficients of operators, and examples.
(Preprint available on arxiv, linked above.) We generalise techniques of Bhat–Skeide (2015) to interpolate comm. families of contractions via comm. families of contractive C₀-semigroups. Building on this, we obtain a continuous non-dilatable family, as an analogue to Parrott's discrete counter-example (1970): For any infinite dimensional Hilbert space H, there is a family of 3 commuting contractive C₀-semigroups, which admits no simultaneous unitary dilation.
This leads to some interesting residuality results within the space of contractive d-parameter contractive C₀-semigroups on a separable ∞-dim. Hilbert space H, where the space is topologised in a certain 'weak' sense:
1) the subset of unitary semigroups is dense G_δ (resp. has meagre and thus nowhere dense closure) if d ∈ {1, 2} (resp. d ≥ 3);
2) the subset of unitarily dilatable and unitarily approximable (resp. non-dilatable and non-approximable) semigroups is residual if d ∈ {1, 2} (resp. d ≥ 3).
Similar residuality results hold for the discrete setting, i.e. the space of comm. families of contractions on H, equipped with the PW-topology. We furthermore obtain the following reduction of Parrott's conjecture:
3) Let d ≥ 3. The subset of d-tuples of (commuting) contractions satisfying the von Neumann inequality is either equal to the whole space, or else is a closed meagre (thus nowhere dense) subset.
In addition we bring both the Bhat–Skeide interpolation and our 0–1 results to bear on the embedding problem. Notably, we show that the Bhat–Skeide interpolation is in a geometric sense optimal, and that generic pairs of commuting contractions can be embedded into commuting pairs of C0-semigroups, which extends Eisner's results (2009–10).
Motivated by two norm equations used to characterize the Friedrichs angle, this paper studies C*-isomorphisms associated with two projections by introducing the matched triple and the semi-harmonious pair of projections. A triple (P, Q, H) is said to be matched if H is a Hilbert C*-module, P and Q are projections on H such that their infimum P ∧ Q exists as an element of \({\cal L}(H)\), where \({\cal L}(H)\) denotes the set of all adjointable operators on H. The C*-subalgebras of \({\cal L}(H)\) generated by elements in {P − P ∧ Q, Q − P ∧ Q, I} and {P, Q, P ∧ Q, I} are denoted by i(P, Q, H) and o(P, Q, H), respectively. It is proved that each faithful representation (π, X) of o(P, Q, H) can induce a faithful representation \((\tilde \pi ,X)\) of i(P, Q, H) such that
When (P, Q) is semi-harmonious, that is, \(\overline {{\cal R}(P + Q)} \) and \(\overline {{\cal R}(2I - P - Q)} \) are both orthogonally complemented in H, it is shown that i(P, Q, H) and i(I − Q, I − P, H) are unitarily equivalent via a unitary operator in \({\cal L}(H)\). A counterexample is constructed, which shows that the same may be not true when (P, Q) fails to be semi-harmonious. Likewise, a counterexample is constructed such that (P, Q) is semi-harmonious, whereas (P, I − Q) is not semi-harmonious. Some additional examples indicating new phenomena of adjointable operators acting on Hilbert C*-modules are also provided.
Conditions for linear integral operators on $L_p$ over measure spaces to satisfy the polynomial covariance type commutation relations are described in terms of defining kernels of the corresponding integral operators. Representation by integral operators are studied both for general polynomial covariance commutation relations and for important classes of polynomial covariance commutation relations associated to arbitrary monomials and to affine functions. Representations of the covariance type commutation relations by integral Volterra operators and integral convolution operators are investigated.
Hilbert modules over a $C^*$-category were first defined by Mitchener, where it was also proved that they form a $C^*$-category. An Eilenberg-Watts theorem for Hilbert modules over $C^*$-algebras was proved by Blecher. We follow a similar path to prove an Eilenberg-Watts theorem for Hilbert modules over $C^*$-categories, characterize equivalences of categories of Hilbert modules as being given by tensoring with imprimitivity bimodules, and employ our results to prove several equivalences of bicategories of $C^*$-algebras and $C^*$-categories.
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