Complex borel structure inJC-algebras

1991 ◽  
Vol 206 (1) ◽  
pp. 225-232 ◽  
Author(s):  
L. J. Bunce ◽  
J. D. Maitland Wright
Keyword(s):  
Borel Structure

scholarly journals The Borel structure of iterates of continuous functions

1989 ◽  
Vol 32 (3) ◽  
pp. 483-494 ◽  
Author(s):  
Paul D. Humke ◽  
M. Laczkovich
Keyword(s):  
Banach Space ◽  
Continuous Functions ◽  
Baire Space ◽  
Space Of Continuous Functions ◽  
Borel Structure ◽  
Image Position ◽  
Set Of Points ◽  
Positive Integers

Let C[0,1] be the Banach space of continuous functions defined on [0,1] and let C be the set of functions f∈C[0,1] mapping [0,1] into itself. If f∈C, fk will denote the kth iterate of f and we put Ck = {fk:f∈C;}. The set of increasing (≡ nondecreasing) and decreasing (≡ nonincreasing) functions in C will be denoted by ℐ and D, respectively. If a function f is defined on an interval I, we let C(f) denote the set of points at which f is locally constant, i.e.We let N denote the set of positive integers and NN denote the Baire space of sequences of positive integers.

scholarly journals CONTINUITY OF UNIVERSALLY MEASURABLE HOMOMORPHISMS

2019 ◽  
Vol 7 ◽  
Author(s):  
CHRISTIAN ROSENDAL
Keyword(s):  
New York ◽  
Functional Analysis ◽  
Chromatic Number ◽  
Set Theory ◽  
Measure Theory ◽  
Polish Groups ◽  
Hamming Graph ◽  
Borel Structure ◽  
North Holland Publishing ◽  
Haar Null Sets

Answering a longstanding problem originating in Christensen’s seminal work on Haar null sets [Math. Scand. 28 (1971), 124–128; Israel J. Math. 13 (1972), 255–260; Topology and Borel Structure. Descriptive Topology and Set Theory with Applications to Functional Analysis and Measure Theory, North-Holland Mathematics Studies, 10 (Notas de Matematica, No. 51). (North-Holland Publishing Co., Amsterdam–London; American Elsevier Publishing Co., Inc., New York, 1974), iii+133 pp], we show that a universally measurable homomorphism between Polish groups is automatically continuous. Using our general analysis of continuity of group homomorphisms, this result is used to calibrate the strength of the existence of a discontinuous homomorphism between Polish groups. In particular, it is shown that, modulo $\text{ZF}+\text{DC}$ , the existence of a discontinuous homomorphism between Polish groups implies that the Hamming graph on $\{0,1\}^{\mathbb{N}}$ has finite chromatic number.

The Classification of Factors is not Smooth

1973 ◽  
Vol 25 (1) ◽  
pp. 96-102 ◽  
Author(s):  
E. J. Woods
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Explicit Construction ◽  
Countable Family ◽  
Borel Sets ◽  
Isomorphism Classes ◽  
Borel Structure

There is a natural Borel structure on the set F of all factors on a separable Hilbert space [3]. Let denote the algebraic isomorphism classes in F together with the quotient Borel structure. Now that various non-denumerable families of mutually non-isomorphic factors are known to exist [1; 6; 8; 10; 11; 12; 13], the most obvious question to be resolved is whether or not is smooth (i.e. is there a countable family of Borel sets which separate points). We answer this question negatively by an explicit construction.

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