scholarly journals THE JIANG–SU ALGEBRA AS A FRAÏSSÉ LIMIT

2017 ◽  
Vol 82 (4) ◽  
pp. 1541-1559 ◽  
Author(s):  
SHUHEI MASUMOTO
Keyword(s):  
Elementary Proof ◽  
Continuous Functions ◽  
C Algebras ◽  
Fraïssé Class

AbstractIn this paper, we give a self-contained and quite elementary proof that the class of all dimension drop algebras together with their distinguished faithful traces forms a Fraïssé class with the Jiang–Su algebra as its limit. We also show that the UHF algebras can be realized as Fraïssé limits of classes of C*-algebras of matrix-valued continuous functions on [0,1] with faithful traces.

scholarly journals The square root of a positive self-adjoint operator

1968 ◽  
Vol 8 (1) ◽  
pp. 17-36 ◽  
Author(s):  
S. J. Bernau
Keyword(s):  
Elementary Proof ◽  
Adjoint Operator ◽  
Continuous Functions ◽  
Spectral Theorem ◽  
Square Root ◽  
Spaces Of Continuous Functions ◽  
Elementary Construction ◽  
Adjoint Operators ◽  
Spectral Family ◽  
Insight Into

One elementary proof of the spectral theorem for bounded self-adjoint operators depends on an elementary construction for the square root of a bounded positive self-adjoint operator. The purpose of this paper is to give an elementary construction for the unbounded case and to deduce the spectral theorem for unbounded self-adjoint operators. In so far as all our results are more or less immediate consequences of the spectral theorem there is little is entirely new. On the other hand the elementary approach seems to the author to provide a deeper insight into the structure of the problem and also leads directly to the spectral theorem without appealing first to the bounded case. Besides this, our methods for proving uniqueness of the square root and of the spectral family seem to be new even in the bounded case. In particular there is no need to invoke representation theorems for linear functionals on spaces of continuous functions.

scholarly journals A Continuous Field of Projectionless C*-Algebras

2001 ◽  
Vol 53 (1) ◽  
pp. 51-72 ◽  
Author(s):  
Andrew Dean
Keyword(s):  
Continuous Functions ◽  
Unit Interval ◽  
Type I ◽  
Crossed Products ◽  
Inductive Limits ◽  
Direct Sums ◽  
Matrix Algebras ◽  
C Algebras ◽  
Finite Dimensional ◽  
Continuous Field

AbstractWe use some results about stable relations to show that some of the simple, stable, projectionless crossed products of O2 by considered by Kishimoto and Kumjian are inductive limits of type C*-algebras. The type I C*-algebras that arise are pullbacks of finite direct sums of matrix algebras over the continuous functions on the unit interval by finite dimensional C*-algebras.

scholarly journals Noncommutative Lattices and the Algebras of Their Continuous Functions

1998 ◽  
Vol 10 (04) ◽  
pp. 439-466 ◽  
Author(s):  
Elisa Ercolessi ◽  
Giovanni Landi ◽  
Paulo Teotonio-Sobrinho
Keyword(s):  
Quantum Physics ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Hausdorff Topology ◽  
Topological Information ◽  
C Algebras ◽  
Finite Dimensional ◽  
Noncommutative Algebras ◽  
Af Algebras ◽  
The Continuum

Recently a new kind of approximation to continuum topological spaces has been introduced, the approximating spaces being partially ordered sets (posets) with a finite or at most a countable number of points. The partial order endows a poset with a nontrivial non-Hausdorff topology. Their ability to reproduce important topological information of the continuum has been the main motivation for their use in quantum physics. Posets are truly noncommutative spaces, or noncommutative lattices, since they can be realized as structure spaces of noncommutative C*-algebras. These noncommutative algebras play the same rôle as the algebra of continuous functions [Formula: see text] on a Hausdorff topological space M and can be thought of as algebras of operator valued functions on posets. In this article, we will review some mathematical results that establish a duality between finite posets and a certain class of C*-algebras. We will see that the algebras in question are all postliminal approximately finite dimensional (AF) algebras.

scholarly journals Hermitian Operators and Isometries on Banach Algebras of Continuous Maps with Values in Unital Commutative C⁎-Algebras

2018 ◽  
Vol 2018 ◽  
pp. 1-14
Author(s):  
Osamu Hatori
Keyword(s):  
Banach Algebras ◽  
Continuous Functions ◽  
C Algebras ◽  
Lipschitz Maps ◽  
Spaces Of Continuous Functions ◽  
Continuous Maps

We study isometries on algebras of the Lipschitz maps and the continuously differentiable maps with the values in a commutative unital C⁎-algebra. A precise proof of a theorem of Jarosz concerning isometries on spaces of continuous functions is exhibited.

CROSSED PRODUCTS OF CERTAIN C*-ALGEBRAS

2014 ◽  
Vol 25 (02) ◽  
pp. 1450010 ◽  
Author(s):  
JIAJIE HUA ◽  
YAN WU
Keyword(s):  
Simple Formula ◽  
Continuous Functions ◽  
Cantor Set ◽  
Crossed Product ◽  
Crossed Products ◽  
C Algebra ◽  
C Algebras ◽  
Tracial Rank ◽  
Universal Coefficient Theorem ◽  
Unital Simple

Let X be a Cantor set, and let A be a unital separable simple amenable [Formula: see text]-stable C*-algebra with rationally tracial rank no more than one, which satisfies the Universal Coefficient Theorem (UCT). We use C(X, A) to denote the algebra of all continuous functions from X to A. Let α be an automorphism on C(X, A). Suppose that C(X, A) is α-simple, [α|1⊗A] = [ id |1⊗A] in KL(1 ⊗ A, C(X, A)), τ(α(1 ⊗ a)) = τ(1 ⊗ a) for all τ ∈ T(C(X, A)) and all a ∈ A, and [Formula: see text] for all u ∈ U(A) (where α‡ and id‡ are homomorphisms from U(C(X, A))/CU(C(X, A)) → U(C(X, A))/CU(C(X, A)) induced by α and id, respectively, and where CU(C(X, A)) is the closure of the subgroup generated by commutators of the unitary group U(C(X, A)) of C(X, A)), then the corresponding crossed product C(X, A) ⋊α ℤ is a unital simple [Formula: see text]-stable C*-algebra with rationally tracial rank no more than one, which satisfies the UCT. Let X be a Cantor set and 𝕋 be the circle. Let γ : X × 𝕋n → X × 𝕋n be a minimal homeomorphism. It is proved that, as long as the cocycles are rotations, the tracial rank of the corresponding crossed product C*-algebra is always no more than one.

scholarly journals Dini's theorem for almost periodic functions

1961 ◽  
Vol 2 (2) ◽  
pp. 143-146 ◽  
Author(s):  
W. Greve
Keyword(s):  
Periodic Function ◽  
Elementary Proof ◽  
Continuous Functions ◽  
Mean Value ◽  
Almost Periodic Function ◽  
Almost Periodic ◽  
Matrix Representations ◽  
Abstract Group ◽  
Compact Topological Space ◽  
Compact Topology

The object of this note is to extend Dini's theorem about (monotonic) sequences of continuous functions on a compact topological space to the case where the underlying domain is an abstract group which is free from topological restrictions. Continuous functions are replaced by almost periodic real-valued functions and the main result may be stated as follows: If a monotonically increasing sequence (fn) of almost periodic real-valued functions on a group G converges pointwise to an almost periodic function f on G, then the sequence converges to f uniformly. The basic idea of the present (elementary) proof is due to v. Kampen [2] and A. Weil [4], i.e., every almost periodic function on a group induces a kind of compact topology in it, relative to which the function is continuous. We modify this idea with the aid of the mean-value of an almost periodic function and obtain a pseudometric topology. This topology facilitates convergence proofs greatly. Moreover, it turns out to be equivalent with the previous one (Lemma 1). No use will be made of the theory of bounded matrix representations. This is significant as any use of the ‘Approximation Theorem’ [3, p. 66, see also p. 226] would violate the claim of an elementary proof.

On the Structure of Finite Level and ω-Decomposable Borel Functions

2013 ◽  
Vol 78 (4) ◽  
pp. 1257-1287 ◽  
Author(s):  
Luca Motto Ros
Keyword(s):  
Banach Space ◽  
Elementary Proof ◽  
Borel Function ◽  
Continuous Functions ◽  
Full Description ◽  
Countable Union ◽  
Space Theory ◽  
Measurable Functions ◽  
Borel Functions

AbstractWe give a full description of the structure under inclusion of all finite level Borel classes of functions, and provide an elementary proof of the well-known fact that not every Borel function can be written as a countable union of Σα0-measurable functions (for every fixed 1 ≤ α < ω1). Moreover, we present some results concerning those Borel functions which are ω-decomposable into continuous functions (also called countably continuous functions in the literature): such results should be viewed as a contribution towards the goal of generalizing a remarkable theorem of Jayne and Rogers to all finite levels, and in fact they allow us to prove some restricted forms of such generalizations. We also analyze finite level Borel functions in terms of composition of simpler functions, and we finally present an application to Banach space theory.

scholarly journals CONTINUOUS SLICE FUNCTIONAL CALCULUS IN QUATERNIONIC HILBERT SPACES

2013 ◽  
Vol 25 (04) ◽  
pp. 1350006 ◽  
Author(s):  
RICCARDO GHILONI ◽  
VALTER MORETTI ◽  
ALESSANDRO PEROTTI
Keyword(s):  
Hilbert Spaces ◽  
Functional Calculus ◽  
Normal Operator ◽  
Regular Function ◽  
Continuous Functions ◽  
Continuous Functional ◽  
Unbounded Operators ◽  
C Algebras ◽  
Normal Operators ◽  
The One

The aim of this work is to define a continuous functional calculus in quaternionic Hilbert spaces, starting from basic issues regarding the notion of spherical spectrum of a normal operator. As properties of the spherical spectrum suggest, the class of continuous functions to consider in this setting is the one of slice quaternionic functions. Slice functions generalize the concept of slice regular function, which comprises power series with quaternionic coefficients on one side and that can be seen as an effective generalization to quaternions of holomorphic functions of one complex variable. The notion of slice function allows to introduce suitable classes of real, complex and quaternionic C*-algebras and to define, on each of these C*-algebras, a functional calculus for quaternionic normal operators. In particular, we establish several versions of the spectral map theorem. Some of the results are proved also for unbounded operators. However, the mentioned continuous functional calculi are defined only for bounded normal operators. Some comments on the physical significance of our work are included.

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