Narrow Results

Let H be a compact quantum group with faithful Haar measure and bounded counit. If α is an action of H on a C*- algebra A, we show that A is nuclear if and only if the fixed-point subalgebra A^α is nuclear. As a consequence H is a nuclear C*-algebra.

Connes' functional formula of the Riemannian distance is generalized to the Lorentzian case using the so-called Lorentzian distance, the d'Alembert operator and the causal functions of a globally-hyperbolic spacetime. As a step of the presented machinery, a proof of the almost-everywhere smoothness of the Lorentzian distance considered as a function of one of the two arguments is given. Afterwards, using a C*-algebra approach, the spacetime causal structure and the Lorentzian distance are generalized into noncommutative structures giving rise to a Lorentzian version of part of Connes' noncommutative geometry. The generalized noncommutative spacetime consists of a direct set of Hilbert spaces and a related class of C*-algebras of operators. In each algebra a convex cone made of self-adjoint elements is selected which generalizes the class of causal functions. The generalized events, called loci, are realized as the elements of the inductive limit of the spaces of the algebraic states on the C*-algebras. A partial-ordering relation between pairs of loci generalizes the causal order relation in spacetime. A generalized Lorentz distance of loci is defined by means of a class of densely-defined operators which play the role of a Lorentzian metric. Specializing back the formalism to the usual globally-hyperbolic spacetime, it is found that compactly-supported probability measures give rise to a non-pointwise extension of the concept of events.

For a flow α on a C*-algebra one defines a symmetry as the group of automorphisms γ such that γαγ^-1 is a cocycle perturbation of α. We propose to define a core of this symmetry, which acts trivially on the set of equivalence classes of KMS state representations, but may act non-trivially on the set of equivalence classes of covariant irreducible representations. In particular this core acts transitively on the set of those which induce faithful representations of the crossed product by α.

The Weyl algebra — the usual C*-algebra employed to model the canonical commutation relations (CCRs), has a well-known defect, in that it has a large number of representations which are not regular and these cannot model physical fields. Here, we construct explicitly a C*-algebra which can reproduce the CCRs of a countably dimensional symplectic space (S, B) and such that its representation set is exactly the full set of regular representations of the CCRs. This construction uses Blackadar's version of infinite tensor products of nonunital C*-algebras, and it produces a "host algebra" (i.e. a generalized group algebra, explained below) for the σ-representation theory of the Abelian group S where σ(·,·) ≔ e^iB(·,·)/2. As an easy application, it then follows that for every regular representation of on a separable Hilbert space, there is a direct integral decomposition of it into irreducible regular representations (a known result).

A coaction δ of a locally compact group G on a C*-algebra A is maximal if a certain natural map from onto is an isomorphism. All dual coactions on full crossed products by group actions are maximal; a discrete coaction is maximal if and only if A is the full cross-sectional algebra of the corresponding Fell bundle. For every nondegenerate coaction of G on A, there is a maximal coaction of G on an extension of A such that the quotient map induces an isomorphism of the crossed products.

Let Γ=ℤ_m * ℤ_n or ℤ * ℤ_n, and let Γ(h) be the subtree consisting of all reduced words starting with any reduced word h ∈ Γ\{e}. We prove that the C*-algebra generated by and the projection P_h onto the subspace ℓ²(Γ(h)) has a unique nontrivial closed ideal ℐ, ℐ is *-isomorphic to , and the quotient algebra is *-isomorphic to either or depending on the last letter of h. We also prove that is a purely infinite, simple C*-algebra if the last letter of h is a generator of ℤ, and that has a unique nontrivial closed ideal if the last letter of h is a generator of ℤ_n; furthermore, is *-isomorphic to and is again a purely infinite, simple C*-algebra. As consequences, all the C*-algebras above have real rank zero, and is nuclear for any h ≠ e.

Let 1 ∈ A ⊂ B be an inclusion of C*-algebras of C*-index-finite type with depth 2. We try to compute the topological stable rank of B (= tsr(B)) when A has topological stable rank one. We show that tsr(B) ≤ 2 when A is a tsr boundedly divisible algebra, in particular, A is a C*-minimal tensor product UHF ⊗ D with tsr(D) = 1. When G is a finite group and α is an action of G on UHF, we know that a crossed product algebra UHF ⋊_α G has topological stable rank less than or equal to two.

These results are affirmative data to a generalization of a question by Blackadar in 1988.

Let G be an inductive limit of finite cyclic groups, and A be a unital simple projectionless C*-algebra with K₁(A) ≅ G and a unique tracial state, as constructed based on dimension drop algebras by Jiang and Su. First, we show that any two aperiodic elements in Aut(A)/WInn(A) are conjugate, where WInn(A) means the subgroup of Aut(A) consisting of automorphisms which are inner in the tracial representation.

In the second part of this paper, we consider a class of unital simple C*-algebras with a unique tracial state which contains the class of unital simple A𝕋-algebras of real rank zero with a unique tracial state. This class is closed under inductive limits and crossed products by actions of ℤ with the Rohlin property. Let A be a TAF-algebra in this class. We show that for any automorphism α of A there exists an automorphism ᾶ of A with the Rohlin property such that ᾶ and α are asymptotically unitarily equivalent. For the proof we use an aperiodic automorphism of the Jiang-Su algebra.

Let be a separable stable and prime C*-algebra. We show that every proper nontrivial ideal in (and hence every proper ideal in which properly contains ) is not stable.

Some consequences are counterexamples naturally occuring, in the non-σ-unital case, to Zhang's Dichotomy as well as the Hjelmborg–Rørdam theory of stability.

We study some reduced free products of C*-algebras with amalgamations. We give sufficient conditions for the positive cone of the K₀ group to be the largest possible. We also give sufficient conditions for simplicity and uniqueness of trace. We use the latter result to give a necessary and sufficient condition for simplicity and uniqueness of trace of the reduced C*-algebras of the Baumslag–Solitar groups BS(m, n).

UHF flows are the flows obtained as inductive limits of flows on full matrix algebras. We will revisit universal UHF flows and give an explicit construction of such flows on a UHF algebra M_k^∞ for any k and also present a characterization of such flows. Those flows are UHF flows whose cocycle perturbations are almost conjugate to themselves.

We investigate KMS states of Fowler's Nica–Toeplitz algebra associated to a compactly aligned product system X over a semigroup P of Hilbert bimodules. This analysis relies on restrictions of these states to the core algebra which satisfy appropriate scaling conditions. The concept of product system of finite type is introduced. If (G, P) is a lattice ordered group and X is a product system of finite type over P satisfying certain coherence properties, we construct KMS_β states of associated to a scalar dynamics from traces on the coefficient algebra of the product system. Our results were motivated by, and generalize some of the results of Laca and Raeburn obtained for the Toeplitz algebra of the affine semigroup over the natural numbers.

Let j be a nonzero integer and let U be a bounded domain. We construct a Fredholm symbol calculus for the C*-algebra generated by the poly-Bergman projection and the operators of multiplication by continuous functions. We define the j-removal boundary in Hilbert spaces of polyanalytic functions and prove that the quotient poly-Toeplitz C*-algebra generated by cosets of poly-Toeplitz operators with continuous symbols is *-isomorphic to the C*-algebra of continuous functions over the j-essential boundary. Unlike the Bergman case, we also show that if j ≠ ±1 then the j-essential boundary coincides with the set of non-isolated points on the boundary.

We put two C*-algebras together in a noncommutative tensor product using quantum group coactions on them and a bicharacter relating the two quantum groups that act. We describe this twisted tensor product in two equivalent ways, based on certain pairs of quantum group representations and based on covariant Hilbert space representations, respectively. We establish basic properties of the twisted tensor product and study some examples.

We consider an extendible endomorphism α of a C*-algebra A. We associate to it a canonical C*-dynamical system (B, β) that extends (A, α) and is "reversible" in the sense that the endomorphism β admits a unique regular transfer operator β_⁎. The theory for (B, β) is analogous to the theory of classic crossed products by automorphisms, and the key idea is to describe the counterparts of classic notions for (B, β) in terms of the initial system (A, α). We apply this idea to study the ideal structure of a non-unital version of the crossed product C*(A, α, J) introduced recently by the author and A. V. Lebedev. This crossed product depends on the choice of an ideal J in (ker α)^⊥, and if J = (ker α)^⊥ it is a modification of Stacey's crossed product that works well with non-injective α's. We provide descriptions of the lattices of ideals in C*(A, α, J) consisting of gauge-invariant ideals and ideals generated by their intersection with A. We investigate conditions under which these lattices coincide with the set of all ideals in C*(A, α, J). In particular, we obtain simplicity criteria that besides minimality of the action require either outerness of powers of α or pointwise quasinilpotence of α.

Let $K$ be a compact metric space and let $\phi :K\to K$ be continuous. We study a $C^{*}$-algebra ${{ℳ}{𝒞}}_{\phi }$ generated by all multiplication operators by continuous functions on $K$ and a composition operator $C_{\phi }$ induced by $\phi$ on a certain $L^2$ space. Let $\gamma =({\gamma }_1,\dots ,{\gamma }_n)$ be a system of proper contractions on $K$. Suppose that ${\gamma }_1,\dots ,{\gamma }_n$ are inverse branches of $\phi$ and $K$ is self-similar. We consider the Hutchinson measure ${\mu }^H$ of $\gamma$ and the $L^2$ space $L^2(K,{\mu }^H)$. Then we show that the $C^{*}$-algebra ${{ℳ}{𝒞}}_{\phi }$ is isomorphic to the $C^{*}$-algebra ${{𝒪}}_{\gamma }(K)$ associated with $\gamma$ under some conditions.

We obtain all Dirichlet spaces ℱ_q, q ∈ ℝ, of holomorphic functions on the unit ball of ℂ^N as weighted symmetric Fock spaces over ℂ^N. We develop the basics of operator theory on these spaces related to shift operators. We do a complete analysis of the effect of q ∈ ℝ in the topics we touch upon. Our approach is concrete and explicit. We use more function theory and reduce many proofs to checking results on diagonal operators on the ℱ_q. We pick out the analytic Hilbert modules from among the ℱ_q. We obtain von Neumann inequalities for row contractions on a Hilbert space with respect to each ℱ_q. We determine the commutants and investigate the almost normality of the shift operators. We prove that the C*-algebras generated by the shift operators on the ℱ_q fit in exact sequences that are in the same Ext class. We identify the groups K₀ and K₁ of the Toeplitz algebras on the ℱ_q arising in K-theory. Radial differential operators are prominent throughout. Some of our results, especially those pertaining to lower negative values of q, are new even for N = 1. Many of our results are valid in the more general weighted symmetric Fock spaces ℱ_b that depend on a weight sequence b.

John Dauns died on June 4, 2009 of cancer in New Orleans at the age of 73. His work on rings and modules is well-known in the algebra community. However, functional analysts working in the area of C*-algebras are likely to know his name from one theorem that is a corollary of results he and I had obtained in work we did in the mid-sixties of the last century ([17, 27, 18]) and which became known as the Dauns-Hofmann Theorem in C*-algebra theory. It has been known in the C*-text book and monograph literature up to the recent one under this name ([9, 20, 21, 39, 41]) since it provides apparently a useful tool that continues to be used in current research (see e.g. [7, 8, 25, 38, 40]). The problem with the historical record of the Dauns–Hofmann Theorem is that it used to be somewhat obscure how it originated and that the full weight of what was proved was not precisely understood for a long time. As John Dauns was deeply, if not subbornly involved in the development of the early phases of the representation of rings, algebras, C*-algebras (and other classes of algebras) by continuous sections in bundles (sometimes called continuous fields) [11–18], and since his work in this area was substantial and contributed much to a local culture of "sectional representation" at Tulane University ([19, 22, 27–34, 43, 44, 48]), I feel that it is justified to attempt a clarification. He can no longer participate himself in such an attempt; nor would he actually protest the occasional lack of acknowledgment were he alive, because that would be contrary to his ever gentle disposition. This small survey is devoted to shedding some additional light on this portion of John Dauns' work in mathematics; it is natural that it should have a personal tenor by a writer remembering his presence and his work as a collaborator.

Let R be a ring with center Z(R) and n be a fixed positive integer. A mapping f : R → R is said to be n-centralizing on a subset S of R if f(x)xⁿ – xⁿ f(x) ∈ Z(R) holds for all x ∈ S. The main result of this paper states that every n-centralizing generalized derivation F on a (n + 1)!-torsion free semiprime ring is n-commuting. Further, we prove that if a generalized derivation F : R → R is n-centralizing on a nonzero left ideal λ, then either R contains a nonzero central ideal or λD(Z) ⊆ Z(R) for some derivation D of R. As an application, n-centralizing generalized derivations of C*-algebras are characterized.

All algebras considered in this paper are commutative with 1. Let baℓ be the category of bounded Archimedean ℓ-algebras. We investigate Dedekind completions and Dedekind complete algebras in baℓ. We give several characterizations for A ∈ baℓ to be Dedekind complete. Also, given A, B ∈ baℓ, we give several characterizations for B to be the Dedekind completion of A. We prove that unlike general Gelfand-Neumark-Stone duality, the duality for Dedekind complete algebras does not require any form of the Stone–Weierstrass Theorem. We show that taking the Dedekind completion is not functorial, but that it is functorial if we restrict our attention to those A ∈ baℓ that are Baer rings. As a consequence of our results, we give a new characterization of when A ∈ baℓ is a C*-algebra. We also show that A is a C*-algebra if and only if A is the inverse limit of an inverse family of clean C*-algebras. We conclude the paper by discussing how to derive Gleason's theorem about projective compact Hausdorff spaces and projective covers of compact Hausdorff spaces from our results.