Ollie
  • Member for 11 years, 2 months
  • Last seen more than 4 years ago
  • Lancaster, UK
Examples of common false beliefs in mathematics
14 votes

Here's a mistake I've seen from students taking a first course in linear analysis. For a vector $g$ in a Hilbert space $H$, it is true that $\langle f,g\rangle=0$ for every $f\in H$ implies $g=0$. ...

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Tomita-Takesaki versus Frobenius: where is the similarity?
13 votes

A low tech (naive?) piece of intuition comes straight from the definition of the modular operator and what happens if one tries to carry it over to finite fields. The nontrivial automorphism $z\...

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What is the Dunford Integral and why is it useful?
12 votes

Since you've answered part of the question, let me elaborate on the Dunford integral. If $f:\Omega\to E$ is weakly measurable and satisfies $\langle x^* ,f\rangle\in L^1(\Omega)$ for all $x^*\in E^*$ ...

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von neumann algebras and measurable spaces
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8 votes

I highly recommend Segal's original paper Equivalences of Measure Spaces [American Journal of Mathematics Vol. 73, No. 2 (1951), pp. 275-313], where he introduced localizable spaces, since this was ...

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Is there an irreducible, noncompact commuting, nonnormal operator, with spectrum strictly continuous?
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6 votes

Essentially I think weighted shifts should be a sufficiently rich class of operators. Consider, for instance, the following example. Take the doubly infinite sequence $$ w_k=\left\{\begin{array}{ll} ...

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Hyperfiniteness of CCR algebra
5 votes

Yes, Araki-Woods showed they're always ITPFI factors and ITPFI factors are hyperfinite. See the following: Araki-Woods, A classification of factors It's a bit of a monster paper, the stuff on CCR ...

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Von Neumann algebra associated to the infinite Cuntz algebra
4 votes

I'm reasonably convinced the algebra generates all of $B(\mathcal{F}(H))$. Note that $1-\sum_{i=1}^\infty s_is_i^*$ converges strongly to the projection $P\_0$ onto $\mathbb{C}$ and similarly the sum $...

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Second quantization of partial isometry
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4 votes

For any contraction $X$ on $H$, the operator $\Gamma(X)$ (in Alain's notation S(X)) is a contraction. It is essentially algebraic to check that a pair of contractions $X,Y$ on $H$ will satisfy $\Gamma(...

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Are the groups $C( \mathbb{R} ; U(n) )$ isomorphic?
2 votes

Here is another proof. The elements of $C(\mathbb{R};U(n)) $ satisfying $f^2=1$ are functions whose values (under the standard representation of U(n)) are self-adjoint unitaries. There are $n+1$ ...

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Arveson index and curvature
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2 votes

The intuition behind Arveson's index is that it's invariant under "small" perturbations of the generator. But this doesn't really make sense mathematically, so the formal definition is that it's ...

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Free, high quality mathematical writing online?
1 votes

Roland Speicher has some nice introductory material for Free Probability (mini course, survey articles etc.) All available at http://www.mast.queensu.ca/~speicher/survey.html

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