scholarly journals Completely bounded mappings and simplicial complex structure in the primitive ideal space of a $C^*$-algebra

2008 ◽  
Vol 361 (03) ◽  
pp. 1397-1427 ◽  
Author(s):  
Robert J. Archbold ◽  
Douglas W. B. Somerset ◽  
Richard M. Timoney
Keyword(s):  
Simplicial Complex ◽  
Complex Structure ◽  
Primitive Ideal ◽  
Ideal Space ◽  
C Algebra

scholarly journals The primitive ideal space of the C*-algebra of the affine semigroup of algebraic integers

2012 ◽  
Vol 154 (1) ◽  
pp. 119-126 ◽  
Author(s):  
SIEGFRIED ECHTERHOFF ◽  
MARCELO LACA
Keyword(s):  
Recent Result ◽  
Orbit Space ◽  
Crossed Product ◽  
Primitive Ideal ◽  
Prime Ideals ◽  
Ideal Space ◽  
C Algebra ◽  
Ring Of Integers ◽  
Affine Semigroup ◽  
Power Set

AbstractThe purpose of this paper is to give a complete description of the primitive ideal space of the C*-algebra [R] associated to the ring of integers R in a number field K in the recent paper [5]. As explained in [5], [R] can be realized as the Toeplitz C*-algebra of the affine semigroup R ⋊ R× over R and as a full corner of a crossed product C0() ⋊ K ⋊ K*, where is a certain adelic space. Therefore Prim([R]) is homeomorphic to the primitive ideal space of this crossed product. Using a recent result of Sierakowski together with the fact that every quasi-orbit for the action of K ⋊ K* on contains at least one point with trivial stabilizer we show that Prim([R]) is homeomorphic to the quasi-orbit space for the action of K ⋊ K* on , which in turn may be identified with the power set of the set of prime ideals of R equipped with the power-cofinite topology.

The Primitive Ideal Space of a C*-Algebra

1974 ◽  
Vol 26 (1) ◽  
pp. 42-49 ◽  
Author(s):  
John Dauns
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Complete Characterization ◽  
Primitive Ideal ◽  
Ideal Space ◽  
Compact Sets ◽  
Locally Compact ◽  
C Algebra ◽  
Weak Star ◽  
Locally Compact Hausdorff Space

The commutative Gelfand-Naimark Theorem says that any commutative C*-algebra A is isomorphic to the ring C0(M, C) of all continuous complex-valued functions tending to zero outside of compact sets of a locally compact Hausdorff space M. A very important part of this theorem is an intrinsic and also a complete characterization of M as exactly the primitive ideal space of A in the hull-kernel (or weak-star) topology. In the non-commutative case, A ≌ Γ0(M, E)—the ring of sections tending to zero outside of compact subsets of a locally compact Hausdorff space M with values in the stalks or fibers E.

scholarly journals THE GLIMM IDEAL SPACE OF A TWO-STEP NILPOTENT LOCALLY COMPACT GROUP

2001 ◽  
Vol 44 (3) ◽  
pp. 505-526 ◽  
Author(s):  
Eberhard Kaniuth ◽  
William Moran
Keyword(s):  
Compact Group ◽  
Sufficient Conditions ◽  
Locally Compact Group ◽  
Primitive Ideal ◽  
Ideal Space ◽  
Locally Compact ◽  
C Algebra ◽  
Primary 22D25 ◽  
Secondary 22D10 ◽  
Necessary And Sufficient

AbstractFor a two-step nilpotent locally compact group $G$, we determine the Glimm ideal space of the group $C^*$-algebra $C^*(G)$ and its topology. This leads to necessary and sufficient conditions for $C^*(G)$ to be quasi-standard. Moreover, some results about the Glimm classes of points in the primitive ideal space $\mathrm{Prim}(C^*(G))$ are obtained.AMS 2000 Mathematics subject classification: Primary 22D25. Secondary 22D10

Minimal primal ideal spaces and norms of inner derivations of tensor products of C*-algebras

1996 ◽  
Vol 119 (2) ◽  
pp. 297-308 ◽  
Author(s):  
Eberhard Kaniuth
Keyword(s):  
Cross Sections ◽  
Dense Subset ◽  
Tensor Products ◽  
The Other ◽  
Primitive Ideal ◽  
Ideal Space ◽  
C Algebra ◽  
C Algebras ◽  
Other Hand ◽  
Full Algebra

An ideal I in a C*-algebra A is called primal if whenever n ≥ 2 and J1,…, Jn are ideals in A with zero product then Jk ⊆ I for at least one k. The topologized space of minimal primal ideals of A, Min-Primal (A), has been extensively studied by Archbold[3]. Very much in the spirit of Fell's work [14] it was shown in [3, theorem 5·3] (see also [5, theorem 3·4]) that if A is quasi-standard, then A is *-isomorphic to a maximal full algebra of cross-sections of Min-Primal (A). Moreover, if A is separable the fibre algebras are primitive throughout a dense subset. On the other hand, the complete regularization of the primitive ideal space of A gives rise to the space of so-called Glimm ideals of A, Glimm (A). It turned out that A is quasi-standard exactly when Min-Primal (A) and Glimm (A) coincide as sets and topologically [5, theorem 3·3].

scholarly journals Aperiodicity and primitive ideals of row-finite k-graphs

2014 ◽  
Vol 25 (03) ◽  
pp. 1450022 ◽  
Author(s):  
Sooran Kang ◽  
David Pask
Keyword(s):  
Primitive Ideal ◽  
Ideal Space ◽  
Gauge Invariant ◽  
C Algebra ◽  
Primitive Ideals ◽  
Spectral Spaces

We describe the primitive ideal space of the C*-algebra of a row-finite k-graph with no sources when every ideal is gauge invariant. We characterize which spectral spaces can occur, and compute the primitive ideal space of two examples. In order to do this we prove some new results on aperiodicity. Our computations indicate that when every ideal is gauge invariant, the primitive ideal space only depends on the 1-skeleton of the k-graph in question.

scholarly journals The prime spectrum and primitive ideal space of a graph C*-algebra

2014 ◽  
Vol 25 (07) ◽  
pp. 1450070
Author(s):  
Gene Abrams ◽  
Mark Tomforde
Keyword(s):  
Disjoint Union ◽  
Satisfying Condition ◽  
Primitive Ideal ◽  
Prime Ideals ◽  
Ideal Space ◽  
Prime Spectrum ◽  
C Algebra ◽  
Primitive Ideals

We describe primitive and prime ideals in the C*-algebra C*(E) of a graph E satisfying Condition (K), together with the topologies on each of these spaces. In particular, we find that primitive ideals correspond to the set of maximal tails disjoint union the set of finite-return vertices, and that prime ideals correspond to the set of clusters of maximal tails disjoint union the set of finite-return vertices.

scholarly journals Some groups with T1 primitive ideal spaces

1985 ◽  
Vol 38 (1) ◽  
pp. 55-64 ◽  
Author(s):  
A. L. Carey ◽  
W. Moran
Keyword(s):  
Normal Subgroup ◽  
Lie Groups ◽  
Lie Group ◽  
Locally Compact Group ◽  
Primitive Ideal ◽  
Ideal Space ◽  
Solvable Lie Groups ◽  
Locally Compact ◽  
Simply Connected ◽  
Connected Lie Group

AbstractLet G be a second countable locally compact group possessing a normal subgroup N with G/N abelian. We prove that if G/N is discrete then G has T1 primitive ideal space if and only if the G-quasiorbits in Prim N are closed. This condition on G-quasiorbits arose in Pukanzky's work on connected and simply connected solvable Lie groups where it is equivalent to the condition of Auslander and Moore that G be type R on N (-nilradical). Using an abstract version of Pukanzky's arguments due to Green and Pedersen we establish that if G is a connected and simply connected Lie group then Prim G is T1 whenever G-quasiorbits in [G, G] are closed.

scholarly journals Reduction of filtered K-theory and a characterization of Cuntz-Krieger algebras

2014 ◽  
Vol 14 (3) ◽  
pp. 570-613 ◽  
Author(s):  
Sara E. Arklint ◽  
Rasmus Bentmann ◽  
Takeshi Katsura
Keyword(s):  
Primitive Ideal ◽  
Ideal Space ◽  
Real Rank ◽  
Real Rank Zero ◽  
Rank Zero ◽  
K Theory

AbstractWe show that filtered K-theory is equivalent to a substantially smaller invariant for all real-rank-zero C*-algebras with certain primitive ideal spaces—including the infinitely many so-called accordion spaces for which filtered K-theory is known to be a complete invariant. As a consequence, we give a characterization of purely infinite Cuntz–Krieger algebras whose primitive ideal space is an accordion space.

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