C-algebra Research Articles

Fredholm-type operators and index

While in [13] we studied classes of Fredholm-type operators defined by the homomorphism Π from L(X) onto the Calkin algebra C(X), X being a Banach space, we study in this paper two classes of Fredholm-type operators defined by the homomorphism π from L(X) onto the algebra C0(X)=L(X)/F0(X), where F0(X) is the ideal of finite rank operators in L(X). Then we define an index for Fredholm-type operators and we show that this new index satisfies similar properties as the usual Fredholm index.

The nuclear dimension of C⁎-algebras associated to topological flows and orientable line foliations

We show that for any locally compact Hausdorff space Y with finite covering dimension and for any continuous flow R↷Y, the resulting crossed product C⁎-algebra C0(Y)⋊R has finite nuclear dimension. This generalizes previous results for free flows, where this was proved using Rokhlin dimension techniques. As an application, we obtain bounds for the nuclear dimension of C⁎-algebras associated to one-dimensional orientable foliations. This result is analogous to the one we obtained earlier for non-free actions of Z. Some novel techniques in our proof include the use of a conditional expectation constructed from the inclusion of a clopen subgroupoid, as well as the introduction of what we call fiberwise groupoid coverings that help us build a link between foliation C⁎-algebras and crossed products.

Classification of 3-dimensional pairs of quadratic forms over a field

By an n-dimensional quadratic form over a field F (F-form) we mean a homogeneous quadratic polynomial in n variables with coefficients in F. We call two pairs of n-dimensional quadratic F-forms (f1,g1), (f2,g2) isomorphic if there exists a nondegenerate linear change of variables taking (f1,g1) to (f2,g2). We give a complete classification (up to isomorphism) of 3-dimensional pairs of forms (f,g) over an arbitrary field F of characteristic different from 2. We consider separately the cases of isotropic form f+tg (which is equivalent to existence of a common zero of f and g), and the anisotropic one. In the first case we make a classification depending on det(f+tg), and in the second case depending on the even Clifford algebra C0(f+tg) and det(f+tg).

Quotient structures, homomorphisms, and partial wreath products in reality-based algebras

Homomorphisms and quotient structures of reality-based algebras are presented and described; related central idempotents and quotient sets are characterized; and the extent to which a homomorphism preserves the given anti-automorphisms of the algebras is determined. The notion of a partial wreath product, an algebraic abstraction of the wedge product of association schemes, is developed. Prototypical results previously obtained for various special cases (including table algebras and commutative C-algebras) are thus uniformly generalized.

Dense Banach subalgebras of the null sequence algebra which do not satisfy a differential seminorm condition

We construct dense Banach subalgebras A of the null sequence algebra c0 which are spectral-invariant but do not satisfy the D1-condition ‖ab‖A≤K(‖a‖∞‖b‖A+‖a‖A‖b‖∞) for all a,b∈A. The sequences in A vanish in a skewed manner with respect to an unbounded function σ:N→[1,∞).

Similarity degree of Fourier algebras

We show that for a locally compact group G, amongst a class which contains amenable and small invariant neighbourhood groups, its Fourier algebra A(G) satisfies a completely bounded version Pisier's similarity property with similarity degree at most 2. Specifically, any completely bounded homomorphism π:A(G)→B(H) admits an invertible S in B(H) for which ‖S‖‖S−1‖≤‖π‖cb2 and S−1π(⋅)S extends to a ⁎-representation of the C*-algebra C0(G). This significantly improves some results due to Brannan and Samei (2010) [5] and Brannan, Daws and Samei (2013) [4]. We also note that A(G) has completely bounded similarity degree 1 if and only if it is completely isomorphic to an operator algebra if and only if G is finite.

A Banach algebra which is generated by idempotents

In this paper we show that the Banach algebra C0(X), where X is a locally compact Hausdorff space, is generated by idempotents if and only if X is totally disconnected.

Tensor products and multipliers of modules L p on locally compact measure spaces

Projective module tensor products and spaces of multipliers (i.e., bounded module morphisms) of the spaces Lp(µ) and Lq(ν) regarded as modules over the algebras C0(Ω) and B(Ω) on a locally compact space Ω are described. Here B(Ω) consists of bounded Borel functions on Ω, µ and ν are regular Borel measures on Ω, 1 ≤ p, q ≤ ∞ in the case of the base algebra B(Ω), and 1 ≤ p, q < ∞ in the case of the base algebra C0(Ω). (Loosely speaking, both the tensor product and the space of multipliers turn out to be yet other modules, which consist of integrable functions and correspond to their own subscripts on L and measures). It is proved and used as an auxiliary tool that, in the case p, q < ∞ (and, generally, only in this case), the replacement of the base algebra C0(Ω) by B(Ω) leaves the tensor products and multipliers intact.

DYNAMICAL SYSTEMS IN GRAPH C*-ALGEBRAS

Let E be a row finite directed graph with no sinks and (XE, σE) the one-sided edge shift space. Then the graph C*-algebra C*(E) contains the commutative algebra C0(XE). Moreover if E is locally finite so that the canonical completely positive map ϕE on C*(E) is well-defined, ϕE|C0(XE) coincides with the *-homomorphism [Formula: see text]. In this paper we first show that if two edge shift spaces (XE, σE) and (XF, σF) are topologically conjugate, there is an isomorphism of C*(E) onto C*(F), and if the graphs are locally finite the isomorphism transforms ϕE|C0(XE) onto ϕF|C0(XF), which has been known for Cuntz–Krieger algebras. Let ht(ϕE) be Voiculescu–Brown topological entropy of ϕE. In case E is finite, it is well-known that the values ht(ϕE), [Formula: see text], hl(E) and hb(E) all coincide, where [Formula: see text] is the AF core of C*(E) and hl(E), hb(E) are the loop, block entropies of E respectively. If E is irreducible and infinite, [Formula: see text] has been known recently, and here we show that [Formula: see text], where Et is the transposed graph of E. Also some dynamical systems related with AF subalgebras [Formula: see text] of [Formula: see text] are examined to prove that [Formula: see text] for each vertex v.

Canonical Dimension Of Orthogonal Groups

We prove Berhuy{Reichstein's conjecture on the canonical dimension of orthogonal groups showing that for any integer n> 1, the canonical dimension of SO2n+1 and of SO2n+2 is equal to n(n + 1)=2. More precisely, for a given (2n + 1)-dimensional quadratic form dened over an arbitrary eld F of characteristic 6 2, we establish a certain property of the correspondences on the orthogonal grassmannian X of n-dimensional totally isotropic subspaces of , provided that the degree over F of any nite splitting eld of is divisible by 2 n ; this property allows us to prove that the function eld of X has the minimal transcendence degree among all generic splitting elds of . 1. Results Let F be an arbitrary eld of characteristic 6 2, a nondegenerate (2n + 1)- dimensional quadratic form over F (with n > 1), X the orthogonal grassmannian of n-dimensional totally isotropic subspaces of . The variety X is projective, smooth, and geometrically connected; dim X = n(n + 1)=2. We write d(X) for the greatest common divisor of the degrees of all closed points on X. In this paper, a eld extension E=F is called a splitting eld of if the Witt index (see (10) for the denition of the Witt index of a quadratic form) of the form E is maximal (i.e., equal to n). Note that a eld extension E=F is a splitting eld of if and only if the set X(E) is nonempty. We write d( ) for the greatest common divisor of the degrees of all nite splitting elds of . Clearly, d( ) = d(X). Moreover, this integer is a power of 2 not exceeding 2 n . The equality d( ) = 2 n holds if, for example, the even Cliord algebra C0( ) of the

Paschke duality for real and graded C-algebras

Paschke duality for real and graded C-algebras

Locally Compact Pro-C*-Algebras

AbstractLet X be a locally compact non-compact Hausdorff topological space. Consider the algebras C(X), Cb(X), C0(X), and C00(X) of respectively arbitrary, bounded, vanishing at infinity, and compactly supported continuous functions on X. Of these, the second and third are C*-algebras, the fourth is a normed algebra, whereas the first is only a topological algebra (it is indeed a pro-C*- algebra). The interesting fact about these algebras is that if one of them is given, the others can be obtained using functional analysis tools. For instance, given the C*-algebra C0(X), one can get the other three algebras by C00(X) = K(C0(X)), Cb(X) = M(C0(X)), C(X) = Γ(K(C0(X))), where the right hand sides are the Pedersen ideal, the multiplier algebra, and the unbounded multiplier algebra of the Pedersen ideal of C0(X), respectively. In this article we consider the possibility of these transitions for general C*-algebras. The difficult part is to start with a pro-C*-algebra A and to construct a C*-algebra A0 such that A = Γ(K(A0)). The pro-C*-algebras for which this is possible are called locally compact and we have characterized them using a concept similar to that of an approximate identity.

Higher Dimensional Cohomology of Weighted Sequence Algebras

AbstractIt is well known thatc0(Z) is amenable and so its global dimension is zero. In this paper we will investigate the cyclic and Hochschild cohomology of Banach algebrac0(Z, ω-1) and its unitisation with coefficients in its dual space, where ω is a weight on Z which satisfies inf {ω(i)} = 0.Moreover we show that the weak homological bi-dimension ofc0(Z, ω-1) is infinity.

Range-inclusive maps on C-algebras

Range-inclusive maps on C-algebras

Hopf C*-Algebras

In this paper we define and study Hopf C*-algebras. Roughly speaking, a Hopf C*-algebra is a C*-algebra A with a comultiplication φ: A → M(A ⊗ A) such that the maps a ⊗ b ↦ φ(a)(1 ⊗ b) and a ↦ (a ⊗ 1)φ(b) have their range in A ⊗ A and are injective after being extended to a larger natural domain, the Haagerup tensor product A ⊗ h A. In a purely algebraic setting, these conditions on φ are closely related to the existence of a counit and antipode. In this topological context, things turn out to be much more subtle, but nevertheless one can show the existence of a suitable counit and antipode under these conditions. The basic example is the C*-algebra C0(G) of continuous complex functions tending to zero at infinity on a locally compact group where the comultiplication is obtained by dualizing the group multiplication. But also the reduced group C*-algebra C r ∗ ( G ) of a locally compact group with the well-known comultiplication falls in this category. In fact all locally compact quantum groups in the sense of Kustermans and the first author (such as the compact and discrete ones) as well as most of the known examples are included. This theory differs from other similar approaches in that there is no Haar measure assumed. 2000 Mathematics Subject Classification: 46L65, 46L07, 46L89.

Transitivity of the norm on Banach spaces¶ having a Jordan structure

We study transitivity conditions on the norm of JB -triples, C-algebras, JB - algebras, and their preduals. We show that, for the predual X of a JBW -triple, each one of the following conditions i) and ii) implies that X is a Hilbert space. i) The closed unit ball of X has some extreme point and the norm of X is convex transitive. ii) The set of all extreme points of the closed unit ball of X is non rare in the unit sphere of X. These results are applied to obtain partial affirmative answers to the open problem whether every JB -triple with transitive norm is a Hilbert space. We extend to arbitrary C-algebras previously known characterizations of transitivity (20) and convex transitivity (36) of the norm on commutative C-algebras. Moreover, we prove that the Calkin algebra has convex transitive norm. We also prove that, if X is a JB -algebra, and if either the norm of X is convex transitive or X has a predual with convex transitive norm, then X is associative. As a consequence, a JB -algebra with almost transitive norm is isomorphic to the field of real numbers.

Spatial numerical ranges of elements of subalgebras of C0(X)

When A is a subalgebra of the commutative Banach algebra C0(X) of all continuous complex‐valued functions on a locally compact Hausdorff space X, the spatial numerical range of element of A can be described in terms of positive measures.

Approximately additive states on type I C-algebras

Approximately additive states on type I C-algebras

Moore Cohomology and Central Twisted Crossed Product C*-Algebras

AbstractLet G be a locally compact second countable group, let X be a locally compact second countable Hausdorff space, and view C(X, T) as a trivial G-module. For G countable discrete abelian, we construct an isomorphism between the Moore cohomology group Hn(G, C(X, T)) and the direct sum Ext(Hn-1(G), Ȟl(βX, Ζ)) ⊕ C(X, Hn(G, T)); here Ȟ1 (βX, Ζ) denotes the first Čech cohomology group of the Stone-Čech compactification of X, βX, with integer coefficients. For more general locally compact second countable groups G, we discuss the relationship between the Moore group H2(G, C(X, T)), the set of exterior equivalence classes of element-wise inner actions of G on the stable continuous trace C*-algebra C0(X) ⊗ 𝒦, and the equivariant Brauer group BrG(X) of Crocker, Kumjian, Raeburn, and Williams. For countable discrete abelian G acting trivially on X, we construct an isomorphism is the group of equivalence classes of principal Ĝ bundles over X first considered by Raeburn and Williams.

The Generalized Even Clifford Algebra

For a central simple algebra A with an orthogonal involution ∗ over a field F of characteristic not equal to two, we associate to it the generalized even Clifford algebraC0(A, ∗). We will present results on the structure of this algebra C0(A, ∗), some of which are due to Jacobson in (J. Algebra1 (1964), 288-300). Using a generic method, we then give a formula for computing C0(A, ∗) when (A, ∗) is decomposable as algebras with involution. Using these results, a necessary condition for decomposability of an algebra with orthogonal involution compatible with the involution is then obtained.