scholarly journals When is the second local multiplier algebra of a C *-algebra equal to the first?

2011 ◽  
Vol 43 (6) ◽  
pp. 1167-1180 ◽  
Author(s):  
Pere Ara ◽  
Martin Mathieu
Keyword(s):  
Multiplier Algebra ◽  
C Algebra

scholarly journals Strict Comparison of Positive Elements in Multiplier Algebras

2017 ◽  
Vol 69 (02) ◽  
pp. 373-407 ◽  
Author(s):  
Victor Kaftal ◽  
Ping Wong Ng ◽  
Shuang Zhang
Keyword(s):  
Positive Element ◽  
Real Rank ◽  
Real Rank Zero ◽  
Multiplier Algebra ◽  
Rank Zero ◽  
C Algebra ◽  
Positive Elements ◽  
Rank One ◽  
Multiplier Algebras ◽  
Positive Linear Combination

AbstractMain result: If a C*-algebrais simple,σ-unital, hasfinitely many extremal traces, and has strict comparison of positive elements by traces, then its multiplier algebraalso has strict comparison of positive elements by traces. The same results holds if finitely many extremal traces is replaced byquasicontinuous scale. A key ingredient in the proof is that every positive element in the multiplier algebra of an arbitrary σ-unital C* -algebra can be approximated by a bi-diagonal series. As an application of strict comparison, ifis a simple separable stable C* -algebra with real rank zero, stable rank one, and strict comparison of positive elements by traces, then whether a positive element is a positive linear combination of projections is determined by the trace values of its range projection.

FRAME-LESS HILBERT C*-MODULES

2018 ◽  
Vol 61 (1) ◽  
pp. 25-31 ◽  
Author(s):  
M. B. ASADI ◽  
M. FRANK ◽  
Z. HASSANPOUR-YAKHDANI
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Compact Operators ◽  
Faithful Representation ◽  
Central Projection ◽  
Multiplier Algebra ◽  
C Algebra ◽  
Infinite Dimensional

AbstractWe show that if A is a compact C*-algebra without identity that has a faithful *-representation in the C*-algebra of all compact operators on a separable Hilbert space and its multiplier algebra admits a minimal central projection p such that pA is infinite-dimensional, then there exists a Hilbert A1-module admitting no frames, where A1 is the unitization of A. In particular, there exists a frame-less Hilbert C*-module over the C*-algebra $K(\ell^2) \dotplus \mathbb{C}I_{\ell^2}$.

On the Structure of Projections and Ideals of Corona Algebras

1989 ◽  
Vol 41 (4) ◽  
pp. 721-742 ◽  
Author(s):  
Shuang Zhang
Keyword(s):  
Separable Hilbert Space ◽  
Affirmative Answer ◽  
Closed Ideal ◽  
Basic Question ◽  
Ideal Structure ◽  
Bounded Operators ◽  
Multiplier Algebra ◽  
C Algebra ◽  
Calkin Algebra ◽  
Nonzero Projection

If K is the set of all compact bounded operators and L(H) is the set of all bounded operators on a separable Hilbert space H, then L(H) is the multiplier algebra of K. In general we denote the multiplier algebra of a C*-algebra A by M(A). For more information about M(A), readers are referred to the articles [1], [3],[7], [9], [14], [18],[20], [23], [26], [27], among others. It is well known that in the Calkin algebra L(H)/K every nonzero projection is infinite. If we assume that A is a-unital (nonunital) and regard the corona algebra M(A)/A as a generalized case of the Calkin algebra, is every nonzero projection in M(A)/A still infinite? Another basic question can be raised: How does the (closed) ideal structure of A relate to the (closed) ideal structure of M(A)/A?In the first part of this note (Sections 1 and 2) we shall give an affirmative answer for the first question if A is a simple a-unital (nonunital) C*-algebra with FS.

scholarly journals The Local Multiplier Algebra of a C*-Algebra, II

2000 ◽  
Vol 171 (2) ◽  
pp. 308-330 ◽  
Author(s):  
D.W.B. Somerset
Keyword(s):  
Multiplier Algebra ◽  
C Algebra ◽  
Algebra Ii

Homotopy Classification of Projections in the Corona Algebra of a Non-simple C*-algebra

2012 ◽  
Vol 64 (4) ◽  
pp. 755-777 ◽  
Author(s):  
Lawrence G. Brown ◽  
Hyun Ho Lee
Keyword(s):  
Hilbert Space ◽  
Compact Operators ◽  
Homotopy Classification ◽  
Multiplier Algebra ◽  
Von Neumann ◽  
C Algebra ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space

AbstractWe study projections in the corona algebra of C(X) ⊗ K, where K is the C*-algebra of compact operators on a separable infinite dimensional Hilbert space and X = [0, 1], [0,∞), (−∞,∞), or [0, 1]/﹛0, 1﹜. Using BDF's essential codimension, we determine conditions for a projection in the corona algebra to be liftable to a projection in the multiplier algebra. We also determine the conditions for two projections to be equal in K0, Murray-von Neumann equivalent, unitarily equivalent, or homotopic. In light of these characterizations, we construct examples showing that the equivalence notions above are all distinct.

AFD MULTIPLIER ALGEBRAS

2006 ◽  
Vol 17 (09) ◽  
pp. 1091-1102
Author(s):  
P. W. NG
Keyword(s):  
Von Neumann Algebra ◽  
Real Rank Zero ◽  
Multiplier Algebra ◽  
Von Neumann ◽  
Rank Zero ◽  
C Algebra ◽  
Unique Trace ◽  
Type Property ◽  
Unital Simple ◽  
Multiplier Algebras

We show that the multiplier algebra of a simple stable nuclear C*-algebra has a property similar to that of an AFD or hyperfinite von Neumann algebra. Specifically, we prove the following: Theorem 0.1. Let [Formula: see text] be a unital simple separable C*-algebra. Let [Formula: see text] be the multiplier algebra of the stabilization of [Formula: see text]. Then [Formula: see text] is nuclear if and only if [Formula: see text] has the AFD-type property. We also study a stronger property called the "strong AFD-type property". We show that if [Formula: see text] is a unital simple real rank zero AT-algebra with unique trace, then the multiplier algebra [Formula: see text] of the stabilization of [Formula: see text] has the strong AFD-type property, and we raise the question of whether this is true more generally.

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