scholarly journals Exotic group C * -algebras of simple Lie groups with real rank one

2021 ◽  
pp. 1-20
Author(s):  
Tim de Laat ◽  
Timo Siebenand
Keyword(s):  
Lie Groups ◽  
Real Rank ◽  
C Algebras ◽  
Simple Lie Groups ◽  
Rank One

scholarly journals Hyperbolic geometry and pointwise ergodic theorems

2017 ◽  
Vol 39 (10) ◽  
pp. 2689-2716
Author(s):  
LEWIS BOWEN ◽  
AMOS NEVO
Keyword(s):  
Large Class ◽  
Lie Groups ◽  
Spectral Theory ◽  
Hyperbolic Space ◽  
Hyperbolic Geometry ◽  
Ergodic Theorems ◽  
Real Rank ◽  
New Approach ◽  
Simple Lie Groups ◽  
Rank One

We establish pointwise ergodic theorems for a large class of natural averages on simple Lie groups of real rank one, going well beyond the radial case considered previously. The proof is based on a new approach to pointwise ergodic theorems, which is independent of spectral theory. Instead, the main new ingredient is the use of direct geometric arguments in hyperbolic space.

Homomorphisms From C(X) Into C*-Algebras

1997 ◽  
Vol 49 (5) ◽  
pp. 963-1009 ◽  
Author(s):  
Huaxin Lin
Keyword(s):  
Real Rank ◽  
Real Rank Zero ◽  
Rank Zero ◽  
C Algebra ◽  
C Algebras ◽  
Finite Dimensional ◽  
Countable Rank ◽  
Rank One ◽  
Dimensional Range

AbstractLet A be a simple C*-algebra with real rank zero, stable rank one and weakly unperforated K0(A) of countable rank. We show that a monomorphism Φ: C(S2) → A can be approximated pointwise by homomorphisms from C(S2) into A with finite dimensional range if and only if certain index vanishes. In particular,we show that every homomorphism ϕ from C(S2) into a UHF-algebra can be approximated pointwise by homomorphisms from C(S2) into the UHF-algebra with finite dimensional range.As an application, we show that if A is a simple C*-algebra of real rank zero and is an inductive limit of matrices over C(S2) then A is an AF-algebra. Similar results for tori are also obtained. Classification of Hom (C(X), A) for lower dimensional spaces is also studied.

scholarly journals Ideal structure and $C^*$-algebras of low rank

2007 ◽  
Vol 100 (1) ◽  
pp. 5 ◽  
Author(s):  
Lawrence G. Brown ◽  
Gert K. Pedersen
Keyword(s):  
Stable Rank ◽  
Low Rank ◽  
Real Rank ◽  
Ideal Structure ◽  
C Algebra ◽  
C Algebras ◽  
Rank One ◽  
The Relationship

We explore various constructions with ideals in a $C^*$-algebra $A$ in relation to the notions of real rank, stable rank and extremal richness. In particular we investigate the maximum ideals of low rank. And we investigate the relationship between existence of infinite or properly infinite projections in an extremally rich $C^*$-algebra and non-existence of ideals or quotients of stable rank one.

GENERALIZED WEYL-VON NEUMANN THEOREMS

1991 ◽  
Vol 02 (06) ◽  
pp. 725-739 ◽  
Author(s):  
HUAXIN LIN
Keyword(s):  
Type Theorem ◽  
Building Blocks ◽  
Real Rank ◽  
Inductive Limits ◽  
Multiplier Algebra ◽  
Direct Sums ◽  
Von Neumann ◽  
C Algebras ◽  
Rank One ◽  
Neumann Type

The conjecture that the real rank of multiplier algebra of every AF-algebra is zero was formally made by L. G. Brown and G. K. Pedersen in [4]. The main purpose of this note is to prove the conjecture. However, we also show that this Weyl-von Neumann type theorem holds for C*-algebras with stable (FU) and stable rank one. In particular, this Weyl-von Neumann type theorem holds for C*-algebras of inductive limits of finite direct sums of basic building blocks with real rank zero and trivial K1-groups considered by George A. Elliott recently.

scholarly journals Prime II1 factors arising from irreducible lattices in products of rank one simple Lie groups

2019 ◽  
Vol 2019 (757) ◽  
pp. 197-246 ◽  
Author(s):  
Daniel Drimbe ◽  
Daniel Hoff ◽  
Adrian Ioana
Keyword(s):  
Lie Groups ◽  
Hyperbolic Groups ◽  
Arithmetic Groups ◽  
Semisimple Lie Groups ◽  
Infinite Groups ◽  
Prime Factorization ◽  
Finite Center ◽  
Simple Lie Groups ◽  
Rank One ◽  
Higher Rank

AbstractWe prove that if Γ is an icc irreducible lattice in a product of connected non-compact rank one simple Lie groups with finite center, then the {\mathrm{II}_{1}} factor {L(\Gamma)} is prime. In particular, we deduce that the {\mathrm{II}_{1}} factors associated to the arithmetic groups {\mathrm{PSL}_{2}(\mathbb{Z}[\sqrt{d}])} and {\mathrm{PSL}_{2}(\mathbb{Z}[S^{-1}])} are prime for any square-free integer {d\geq 2} with {d\not\equiv 1~{}(\operatorname{mod}\,4)} and any finite non-empty set of primes S. This provides the first examples of prime {\mathrm{II}_{1}} factors arising from lattices in higher rank semisimple Lie groups. More generally, we describe all tensor product decompositions of {L(\Gamma)} for icc countable groups Γ that are measure equivalent to a product of non-elementary hyperbolic groups. In particular, we show that {L(\Gamma)} is prime, unless Γ is a product of infinite groups, in which case we prove a unique prime factorization result for {L(\Gamma)}.

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