Trivial endomorphisms of the Calkin algebra

Linear Maps on C*-Algebras Preserving the Set of Operators that are Invertible in

Canadian Mathematical Bulletin ◽

10.4153/cmb-2010-087-x ◽

2011 ◽

Vol 54 (1) ◽

pp. 141-146

Author(s):

Sang Og Kim ◽

Choonkil Park

Keyword(s):

Real Rank ◽

Real Rank Zero ◽

Rank Zero ◽

Calkin Algebra ◽

C Algebras ◽

Linear Maps

AbstractFor C*-algebras of real rank zero, we describe linear maps ϕ on that are surjective up to ideals , and π(A) is invertible in if and only if π(ϕ(A)) is invertible in , where A ∈ and π : → is the quotient map. We also consider similar linear maps preserving zero products on the Calkin algebra.

Download Full-text

Joint Mean Oscillation and Local Ideals in the Toeplitz Algebra II: Local Commutivity and Essential Commutant

Canadian Mathematical Bulletin ◽

10.4153/cmb-2002-034-9 ◽

2002 ◽

Vol 45 (2) ◽

pp. 309-318

Author(s):

Jingbo Xia

Keyword(s):

Local Analysis ◽

Toeplitz Algebra ◽

Calkin Algebra ◽

Algebra Ii ◽

Mean Oscillation ◽

Essential Commutant ◽

Double Commutant ◽

Local Ideals

AbstractA well-known theorem of Sarason [11] asserts that if [Tf, Th] is compact for every h ∈ H∞, then f ∈ H∞ + C(T). Using local analysis in the full Toeplitz algebra τ = τ(L∞), we show that the membership f ∈ H∞ + C(T) can be inferred from the compactness of a much smaller collection of commutators [Tf, Th]. Using this strengthened result and a theorem of Davidson [2], we construct a proper C*-subalgebra τ(L)) of τ which has the same essential commutant as that of τ. Thus the image of τ(ℒ) in the Calkin algebra does not satisfy the double commutant relation [12], [1]. We will also show that no separable subalgebra Ѕ of τ is capable of conferring the membership f ∈ H∞ + C(T) through the compactness of the commutators {[Tf, S] : S ∈ Ѕ}.

Download Full-text

Operators with Compact Self-Commutator

Canadian Journal of Mathematics ◽

10.4153/cjm-1974-012-2 ◽

1974 ◽

Vol 26 (1) ◽

pp. 115-120 ◽

Cited By ~ 1

Author(s):

Carl Pearcy ◽

Norberto Salinas

Keyword(s):

Hilbert Space ◽

Compact Operators ◽

Complex Hilbert Space ◽

Linear Operators ◽

Bounded Linear Operators ◽

Open Problems ◽

Bounded Linear ◽

Infinite Dimensional ◽

Calkin Algebra ◽

The Ideal

Let be a fixed separable, infinite dimensional complex Hilbert space, and let () denote the algebra of all (bounded, linear) operators on . The ideal of all compact operators on will be denoted by and the canonical quotient map from () onto the Calkin algebra ()/ will be denoted by π.Some open problems in the theory of extensions of C*-algebras (cf. [1]) have recently motivated an increasing interest in the class of all operators in () whose self-commuta tor is compact.

Download Full-text

Filters in C*-Algebras

Canadian Journal of Mathematics ◽

10.4153/cjm-2011-095-4 ◽

2013 ◽

Vol 65 (3) ◽

pp. 485-509 ◽

Cited By ~ 5

Author(s):

Tristan Matthew Bice

Keyword(s):

Unit Ball ◽

Pure State ◽

Positive Answer ◽

The State ◽

Natural Numbers ◽

Cardinal Invariant ◽

Calkin Algebra ◽

C Algebras ◽

Positive Elements ◽

Singer Conjecture

AbstractIn this paper we analyze states on C*-algebras and their relationship to filter-like structures of projections and positive elements in the unit ball. After developing the basic theory we use this to investigate the Kadison–Singer conjecture, proving its equivalence to an apparently quite weak paving conjecture and the existence of unique maximal centred extensions of projections coming from ultrafilters on the natural numbers. We then prove that Reid's positive answer to this for q-points in fact also holds for rapid p-points, and that maximal centred filters are obtained in this case. We then show that consistently, such maximal centred filters do not exist at all meaning that, for every pure state on the Calkin algebra, there exists a pair of projections on which the state is 1, even though the state is bounded strictly below 1 for projections below this pair. Next, we investigate towers, using cardinal invariant equalities to construct towers on the natural numbers that do and do not remain towers when canonically embedded into the Calkin algebra. Finally, we show that consistently, all towers on the natural numbers remain towers under this embedding.

Download Full-text