Questions tagged [nuclear-spaces]

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Smooth and decaying matrix coefficients

I want to talk about the matrix coefficients $$ f(g) = \langle \pi(g) v, \alpha^* \rangle.$$ Here $(\pi, V_\pi)$ is a representation of a Lie group $G$, with $V_\pi$ being a topological vector space ...
3 votes
0 answers
76 views

"Weakly" nuclear operators (terminology)

Recently, I'd come across the following kind of operators and I wonder if they have been considered before and given a name. Let's say that a linear map $T:V\to W$ between locally convex topological ...
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4 votes
1 answer
113 views

If $F$ is a countably normed, nuclear Fréchet space, can I then find a fundamental system which exhibits both of these properties at once?

Let $F$ a Fréchet space. This means that $F$ is a complete Hausdorff topological space whose topology can be generated by an increasing family of seminorms $\{ p_{n} \}_{n \in \mathbb{N}}$. Let's ...
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4 votes
1 answer
138 views

Are nuclear spaces used in creating variant theories of distributions?

Laurent Schwartz proved his Kernel Theorem in 1952  to justify extending his theory of distributions to several variables. Then he and Jean Dieudonne gave Alexander Grothendieck the assignment to ...
18 votes
2 answers
684 views

What is known about the "unitary group" of a rigged Hilbert space?

Suppose that $(E,H)$ is a rigged (infinite dimensional, separable) Hilbert space, i.e. $H$ is a Hilbert space, and $E$ is a Fréchet space, equipped with a continuous linear injection $E \rightarrow H$ ...
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6 votes
3 answers
827 views

Are nuclear operators closed under extensions?

Given $X_i, Y_i$ Banach spaces, $f_j, g_j, T_i$ bounded linear operators for $i=1,2,3$ and $j=1,2$. We have the following diagram $\require{AMScd}$ \begin{CD} 0 @>>> X_1 @>f_1>> X_2 ...
9 votes
1 answer
205 views

A non nuclear $C^*$ algebra $A$ for which the algebraic tensor product $A\otimes A$ admits a unique $C^*$ norm

Is there a non nuclear $C^*$ algebra $A$ for which the minimum and maximum $C^*$ norms on $A\otimes A$ coincide? A somewhat similar question is discussed here.
3 votes
1 answer
327 views

multiplier algebra of a simple $C^*$ algebra

If $A=K(H)$, where $H$ is an infinite dimensional separable Hilbert space, then $A$ is simple and nuclear, and the multiplier algebra $M(A)$ of $A$ is not nuclear. My question is: can we find a non-...
1 vote
1 answer
150 views

Nuclear operators and their eigenvalues

I know that if an operator (on a Banach space with approximation property) is nuclear of order zero, then its eigenvalues are $p$-summable for any $p>0$. (I read it from Grothendieck’s book “...
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7 votes
1 answer
250 views

Completely bounded maps approximately factoring through finite matrices

Let $A$, $B$ be two $C^\ast$-algebras and $\mathcal{F}(A,B)$ be the operator ideal of all completely bounded operators $T:A \to B$ for which there are uniformly bounded nets of completely bounded maps ...
2 votes
0 answers
65 views

Nuclear operator between general topological modules over ultrametric Banach rings

In the celebrating paper "Completely continuous endomorphisms of p-adic Banach spaces", Serre established a Fredholm-Riesz theory for compact endomorphisms of Banach spaces over (spherically complete)...
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9 votes
2 answers
424 views

Traces of operators in nuclear spaces

I am currently reading up on nuclear spaces in Jarchow, "Locally Convex Spaces", but I got confused and don't seem to find my mistake. In said book, theorem 21.5.9 states: Let $F$ be a nuclear ...
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4 votes
1 answer
296 views

Relation between dual of nuclear space $(\substack{\text{lim} \\ \leftarrow i} H_i)'$ and $\substack{\text{colim} \\ i \rightarrow } H_i$

Let $\substack{\text{lim} \\ \leftarrow i} H_i$ be a nuclear space, considered as the limit of the codirected diagram $$... \to H_2 \to H_1 \to H_0,$$ with $f_{ji}:H_i \to H_j$ being the trace class ...
8 votes
1 answer
459 views

Is the space $S'(\mathbb{N})$ of slowly increasing sequences the projective limit of Hilbert sequence spaces?

Let $S(\mathbb{N})$ be the space of rapidly decreasing sequences and $S'(\mathbb{N})$ its topological dual, the space of sequences bounded by a polynomial. For $m\in \mathbb{Z}$, we also define $\...
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13 votes
1 answer
1k views

Error in Maurins proof for the nuclear spectral theorem?

I am currently studying the nuclear spectral theorem as presented in K. Maurins Monograph "General Eigenfunction Expansions and Unitary Representations of Topological Groups", second chapter or ...
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3 votes
2 answers
332 views

Linear operators on distributions with different topologies

Denote by $\mathscr{D}^\prime$ and $\mathscr{D}^\prime_b$ the space of distributions on $\mathbb{R}^n$ equipped with the weak and the strong topology, respectively. Because the topology of $\mathscr{D}...
1 vote
0 answers
188 views

Reference for a General Theory of Sequences?

Since decades, mathematicians are studying function spaces, discovering new structures more and more adapted for a general theory of functional analysis. In that works, sequence spaces are generally ...
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10 votes
2 answers
1k views

Wiener measure and Bochner Minlos

I am reading probability theory and I have a question. The Bochner-Minlos theorem roughly says that if we have $E \subset H \subset E^*$ where $H$ is a Hilbert space, then there is a unique measure ...
10 votes
3 answers
698 views

Measure theory in nuclear spaces

Much of the literature on measure theory in linear spaces focuses on the case of normed linear spaces (e.g., the outstanding book by Vakhania, or its sequel). However, nuclear linear spaces "as far ...
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6 votes
2 answers
2k views

What functions can be obtained as a convolution of a Schwartz function and a tempered distribution?

Let $\mathcal S (\mathbb R)$ denote the space of Schwartz functions on $\mathbb R$ and $\mathcal S^* (\mathbb R)$ denote the dual space of Schwartz (a.k.a tempered) distributions. We consider $\...
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18 votes
7 answers
6k views

Grothendieck on Topological Vector Spaces

In a short biography article on Alexander Grothendieck, it is mentioned that after Grothendieck submitted his first thesis on Topological Vector Spaces (TVS), apparently, he told Bernard Malgrange ...
5 votes
1 answer
740 views

Schwartz Kernel theorem for tempred functions

Let $T(R)$ denote the space of tempered functions on the line, i.e. the smooth functions that give Schwartz function after a multiplication by any Schwartz function, equipped with the natural nuclear ...
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8 votes
1 answer
813 views

Exactness of completed tensor product of nuclear spaces

Let $0 \to V \to W \to L \to 0$ be a strict short exact sequence of (complete) nuclear spaces, i.e. it is a short exact sequence of (complete) nuclear spaces, all the maps are continuous, the map $...
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2 votes
0 answers
303 views

Hom of Nuclear spaces

Let V be a Nuclear space (not necessary Frechet, but complete). Let V^* be the dual space considered with the weak topology (i.e. pointwise convergence). Is it true that $V^*$ is also nuclear? Is it ...
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16 votes
0 answers
998 views

Is the category of smooth manifolds equivalent to the opposite category of the category of commutative monoids of some additive symmetric monoidal category?

This is a followup to my previous question, which asked whether the category of commutative or noncommutative C*-algebras or von Neumann algebras is equivalent to the category of commutative or ...
15 votes
2 answers
717 views

Are extensions of nuclear Fréchet spaces nuclear?

Consider the category of Fréchet spaces, the morphisms being continuous linear maps with closed image. Suppose that we have a short exact sequence in that category: $0 \rightarrow V_1 \rightarrow ...
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