On derivations and elementary operators on C*-algebras

2013 ◽  
Vol 56 (2) ◽  
pp. 515-534 ◽  
Author(s):  
Ilja Gogić
Keyword(s):  
Vector Bundle ◽  
Finite Type ◽  
Maximal Ideal ◽  
Operator Norm ◽  
Ideal Space ◽  
Banach Module ◽  
C Algebras ◽  
Constant Dimension ◽  
Elementary Operators ◽  
Bounded Norm

AbstractLet A be a unital C*-algebra with the canonical (H) C*-bundle $\mathfrak{A}$ over the maximal ideal space of the centre of A, and let E(A) be the set of all elementary operators on A. We consider derivations on A which lie in the completely bounded norm closure of E(A), and show that such derivations are necessarily inner in the case when each fibre of $\mathfrak{A}$ is a prime C*-algebra. We also consider separable C*-algebras A for which $\mathfrak{A}$ is an (F) bundle. For these C*-algebras we show that the following conditions are equivalent: E(A) is closed in the operator norm; A as a Banach module over its centre is topologically finitely generated; fibres of $\mathfrak{A}$ have uniformly finite dimensions, and each restriction bundle of $\mathfrak{A}$ over a set where its fibres are of constant dimension is of finite type as a vector bundle.

scholarly journals Sectional representation of Banach modules and their multipliers

2003 ◽  
Vol 2003 (13) ◽  
pp. 817-825
Author(s):  
Terje Hõim ◽  
D. A. Robbins
Keyword(s):  
Banach Algebra ◽  
Maximal Ideal ◽  
Commutative Banach Algebra ◽  
Maximal Ideal Space ◽  
Ideal Space ◽  
Banach Module ◽  
The Past ◽  
Banach Modules ◽  
Quotient Modules

LetXbe a Banach module over the commutative Banach algebraAwith maximal ideal spaceΔ. We show that there is a norm-decreasing representation ofXas a space of bounded sections in a Banach bundleπ:ℰ→Δ, whose fibers are quotient modules ofX. There is also a representation ofM(X), the space of multipliersT:A→X, as a space of sections in the same bundle, but this representation may not be continuous. These sectional representations subsume results of various authors over the past three decades.

scholarly journals On C*-algebras with the approximate n-th root property

2005 ◽  
Vol 72 (2) ◽  
pp. 197-212 ◽  
Author(s):  
A. Chigogidze ◽  
A. Karasev ◽  
K. Kawamura ◽  
V. Valov
Keyword(s):  
Maximal Ideal ◽  
Maximal Ideal Space ◽  
Ideal Space ◽  
Square Root ◽  
C Algebras ◽  
Cech Cohomology

We say that a C*-algebra X has the approximate n-th root property (n ≥ 2) if for every a ∈ X with ∥a∥ ≤ 1 and every ɛ > 0 there exits b ∈ X such that ∥b∥ ≤ 1 and ∥a − bn∥ < ɛ. Some properties of commutative and non-commutative C*-algebras having the approximate n-th root property are investigated. In particular, it is shown that there exists a non-commutative (respectively, commutative) separable unital C*-algebra X such that any other (commutative) separable unital C*-algebra is a quotient of X. Also we illustrate a commutative C*-algebra, each element of which has a square root such that its maximal ideal space has infinitely generated first Čech cohomology.

Elementary operators and subhomogeneous C*-algebras

2010 ◽  
Vol 54 (1) ◽  
pp. 99-111 ◽  
Author(s):  
Ilja Gogić
Keyword(s):  
Tensor Product ◽  
Finite Type ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
C Algebras ◽  
Direct Sum ◽  
Elementary Operators ◽  
Uniformly Bounded

AbstractLet A be a C*-algebra and let ΘA be the canonical contraction form the Haagerup tensor product of M(A) with itself to the space of completely bounded maps on A. In this paper we consider the following conditions on A: (a) A is a finitely generated module over the centre of M(A); (b) the image of ΘA is equal to the set E(A) of all elementary operators on A; and (c) the lengths of elementary operators on A are uniformly bounded. We show that A satisfies (a) if and only if it is a finite direct sum of unital homogeneous C*-algebras. We also show that if a separable A satisfies (b) or (c), then A is necessarily subhomogeneous and the C*-bundles corresponding to the homogeneous subquotients of A must be of finite type.

A non-commutative Gel'fand-Naimark theorem

1980 ◽  
Vol 88 (3) ◽  
pp. 425-428 ◽  
Author(s):  
Christopher J. Mulvey
Keyword(s):  
Maximal Ideal ◽  
Maximal Ideal Space ◽  
Ideal Space ◽  
Isometric Isomorphism ◽  
C Algebra ◽  
C Algebras ◽  
Image Position ◽  
Straightforward Proof

This paper presents a straightforward proof of the Gel'fand-Naimark theorem for non-commutative C*-algebras with identity, established by Dauns and Hofmann(2) in the context of fields of C*-algebras, by considering instead C*-algebras in categories of sheaves. The proof differs from that of (2,3,4) in obtaining an isometric *-isomorphismfrom the C*-algebra A to the C*-algebra of sections of a C*-algebra Ax in the category of sheaves on the maximal ideal space X of the centre of A, without invoking any arguments which involve completeness (3, Theorem 7·9). Instead, the results of (7) yield immediately the existence of an algebraic isomorphism, the compactness of the maximal ideal space X then being used to prove that Ax is indeed a C*-algebra in the category of sheaves on X and that the isomorphism is isometric. One recovers the representation of (2) by noting (8) that any C*-algebra in the category of sheaves on X is isomorphic to the sheaf of sections of a canonical field of C*-algebras on X.

scholarly journals The convolution algebraH1(R)

2010 ◽  
Vol 8 (2) ◽  
pp. 167-179 ◽  
Author(s):  
R. L. Johnson ◽  
C. R. Warner
Keyword(s):  
Banach Algebra ◽  
Maximal Ideal ◽  
Tauberian Theorem ◽  
Singular Integrals ◽  
Approximate Identity ◽  
Maximal Ideal Space ◽  
Ideal Space

H1(R) is a Banach algebra which has better mapping properties under singular integrals thanL1(R) . We show that its approximate identity sequences are unbounded by constructing one unbounded approximate identity sequence {vn}. We introduce a Banach algebraQthat properly lies betweenH1andL1, and use it to show thatc(1 + lnn) ≤ ||vn||H1≤Cn1/2. We identify the maximal ideal space ofH1and give the appropriate version of Wiener's Tauberian theorem.

scholarly journals Diameter-preserving linear bijections of function spaces

2001 ◽  
Vol 70 (3) ◽  
pp. 323-336 ◽  
Author(s):  
T. S. S. R. K. Rao ◽  
A. K. Roy
Keyword(s):  
Maximal Ideal ◽  
Convex Set ◽  
Compact Convex ◽  
Extreme Points ◽  
Continuous Functions ◽  
Ideal Space ◽  
Function Algebras ◽  
Compact Convex Set ◽  
Vector Valued Functions ◽  
Vector Valued

AbstractIn this paper we give a complete description of diameter-preserving linear bijections on the space of affine continuous functions on a compact convex set whose extreme points are split faces. We also give a description of such maps on function algebras considered on their maximal ideal space. We formulate and prove similar results for spaces of vector-valued functions.

Quasi-standard C*-algebras

1990 ◽  
Vol 107 (2) ◽  
pp. 349-360 ◽  
Author(s):  
R. J. Archbold ◽  
D. W. B. Somerset
Keyword(s):  
Cross Sections ◽  
Equivalence Relation ◽  
Dense Subset ◽  
Base Space ◽  
Ideal Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
C Algebras ◽  
Necessary And Sufficient ◽  
Full Algebra

AbstactA necessary and sufficient condition is given for a separable C*-algebra to be *-isomorphic to a maximal full algebra of cross-sections over a base space such that the fibre algebras are primitive throughout a dense subset. The condition is that the relation of inseparability for pairs of points in the primitive ideal space should be an open equivalence relation.

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