Questions tagged [tensor-products]

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Clebsch–Gordan decomposition for $\mathrm{SU}(2)$, in indices

Let $\pi_m$, $m \geq 0$, be the unitary irreps of $\mathrm{SU}(2)$. The Clebsch–Gordan decomposition then gives that $$ \pi_m \otimes \pi_n = \bigoplus_{k=0}^{\min(m,n)}\pi_{m+n-2k}.$$ But suppose I ...
onamoonlessnight's user avatar
1 vote
1 answer
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The tensor product of two bounded operators

Let $E$, $F$ be two complex Hilbert spaces and $\mathcal{L}(E)$ (resp. $\mathcal{L}(F)$) be the algebra of all bounded linear operators on $E$ (resp. $F$). The algebraic tensor product of $E$ and $F$ ...
Student's user avatar
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Description of (completely) bounded operator

I am somewhat a beginner in the field of operator algebras and was wondering about the following: Let $T$ be a linear map between the space of bounded operators $B(H)$ on some Hilbert space and $S$ a ...
Hörmander123's user avatar
5 votes
1 answer
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Approximation property counterexamples? (Also: relation to tensor products)

I remember reading somewhere (but unfortunately, I've forgotten where it was) that the canonical map from the (completed) projective tensor product of two Banach spaces to the (completed) injective ...
Jeff Egger's user avatar
2 votes
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156 views

A formula for vector valued measurable functions

Let $B_{\infty}(\Omega)$ be the space of bounded measurable functions on the measurable space $\Omega$. For a given Banach space $X$, let us denote $B_{\infty}(\Omega,X)$ by the set of all bounded ...
ABB's user avatar
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Reference request: tensor induction

While working on a problem, I constructed something which looked like an induced representation, but with a tensor product instead of a direct sum. Here is a special case. Let $G$ be a group, with ...
D_S's user avatar
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1 answer
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On ranks of matrices with tensor structure

Fix two $2^t$ length vector of form $p=\begin{bmatrix}u_1&v_1\end{bmatrix}\otimes\dots\otimes\begin{bmatrix}u_t&v_t\end{bmatrix}$ and $r=\begin{bmatrix}w_1&z_1\end{bmatrix}\otimes\dots\...
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The existence of $v\in A\otimes_{\mathbb{K}}A$ such that $(a\otimes_{\mathbb{K}}1)v=(1\otimes_{\mathbb{K}}a)v$

If $A$ is a finite dimensional commutative, associative, unital algebra over a field $\mathbb{K}$ then does there exist a non-zero vector $v\in A\otimes_{\mathbb{K}}A$ such that $(a\otimes_{\mathbb{K}}...
Campbell's user avatar
8 votes
0 answers
331 views

Torsion in a tensor product over a group ring

Let $\Gamma$ be a finitely generated dense subgroup of a pro-$p$ group $G$. Let $\mathbb Z_p$ be the ring of $p$-adic numbers. Denote by $\mathbb Z_p[[G]]$ the completed group algebra. Is it true ...
Andrei Jaikin's user avatar
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1 answer
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Is the canonical map $\mathfrak L(X,E)\:\hat\otimes_\pi\:\mathfrak L(Y,F)\to\mathfrak L(X\:\hat\otimes_\pi\:Y,E\:\hat\otimes_\pi\:F)$ injective?

If $A,B$ are $\mathbb R$-Banach spaces, let $A\:\hat\otimes_\pi\:B$ denote the completion of the algebraic tensor product of $A$ and $B$ with respect to the projective norm. Let $X,Y,E,F$ be $\mathbb ...
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If $H$ is a Hilbert space, is the projective tensor product $E\:\hat\otimes_\pi\:H$ isometrically isomorphic to $E\:\hat\otimes_\pi\:H'$?

Let $E$ be a $\mathbb R$-Banach space $H$ be a $\mathbb R$-Hilbert space $E\:\hat\otimes_\pi\:H$ denote the completion of the tensor product of $E$ and $H$ with respect to the projective norm By ...
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Is there a vector-valued trace such that $\text{tr}((L\otimes_π\text{id}_H)T)=LT$ for all $L∈\mathfrak L(H,\mathfrak L(H))$ and $T∈H\hat\otimes_πH$?

Let $H$ be a separable $\mathbb R$-Hilbert space $L\in\mathfrak L(H,\mathfrak L(H,\mathbb R))$ $T\in\mathfrak L(H)$ be nonnegative, self-adjoint and nuclear (trace-class) Note that$^1$ $$\...
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Strassen-like algorithm for Hadamard product of $2\times 2$ matrices

Let $A=\pmatrix{a_1& a_2\\a_3&a_4}, B=\pmatrix{b_1& b_2\\b_3&b_4}$ be two matrices. Let $C$ be the Hadamard product of $A$ and $B$. $$C=\pmatrix{c_1& c_2\\c_3&c_4}=\pmatrix{...
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If $f$ takes values in $L(H,L(H,\Bbb R))$ and $μ$ is a $H\hat ⊗_πH$-valued measure, how are $\int f\:dμ$ and $\int f⊗_π\text{id}_Hdμ$ related?

Let $H$ be a separable $\mathbb R$-Hilbert space $H\:\hat\otimes_\pi\:H$ denote the projective tensor product of $H$ and $H$ $(\Omega,\mathcal A)$ be a measurable space $\mu$ be a $H\:\hat\otimes_\pi\...
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Tensor product and quotients of it [closed]

Let $A,B$ be Banach algebras, and $I$ be a closed two sided ideal of $A$ and $J$ be a closed two sided ideal of $B$. Is the relation $A\hat{\otimes}B/I\hat{\otimes}J\cong A/I\hat{\otimes}B/J$ true?(...
Albert harold's user avatar
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Exterior tensor product of $D$ Modules

The exterior tensor product of sheaves of modules is defined as: $M \boxtimes N = p_1^{*}M \otimes_{\mathcal{O}_{X \times Y}} p_2^{*}N \cong \mathcal{O}_{X \times Y} \otimes_{p_1^{-1}\mathcal{O}_X \...
Federico Barbacovi's user avatar
4 votes
1 answer
327 views

Tensor sum of two operators

Let $E$ be a complex Hilbert space. Let $E\overline{\otimes}E$ denotes the completion, endowed with a reasonable uniform cross-norm of the algebraic tensor product $E\otimes E$. Definition: Let $A,...
Student's user avatar
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torsion free modules $M$ over Noetherian domain of dimension $1$ for which $l(M/aM) \le (\dim_K K \otimes_R M) \cdot l(R/aR), \forall 0 \ne a \in R$

Let $R$ be a Noetherian domain of Krull-dimension $1$ (i.e. every non-zero prime ideal maximal). Let $M$ be a torsion free $R$-module . Let $K$ be the fraction-field of $R$ and let $r=\dim_K S^{-1}M=\...
user avatar
3 votes
1 answer
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Algebraic tensor product of C*-algebras extends via ideals? Application to restriction theorem?

Is the following assertion and the proof below correct, or am I missing something very important? Moreover, would the corollaries be correct then? Besides, I would also appreciate a lot any comment, ...
C-star-W-star's user avatar
1 vote
1 answer
135 views

Proving the hyponormality of $A\otimes B$

Let $E$ be a complex Hilbert space. We recall that an operator $T\in\mathcal{L}(E)$ is said to be hyponormal if $[T^*, T]\geq 0$ (i.e. $\langle (T^*T-TT^*)x,x \rangle\geq 0$ for all $x\in E$). Let $E\...
Student's user avatar
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3 votes
1 answer
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Cohomology of a homotopy pullback of groupoids

Let $\Lambda \stackrel{F}{\to} \Omega \stackrel{G}{\leftarrow} \Gamma$ be a diagram of groupoids and functors and $\Gamma \times_\Omega \Lambda$ the homotopy pullback. We will regard all these ...
Lukas Woike's user avatar
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4 votes
1 answer
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approximate diagonal

Let $I$ be an arbitary index set, $((A_i)_i,\|.\|_i)_{i\in I}$ be a family of Banach algebras, with approximate diagonal $(m^{i}_α)_α\subseteq A_i\hat\otimes A_i$, and $B=\{(x_i)_i\in {\displaystyle \...
R.N's user avatar
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Projective tensor product

Let $A$ and $B$ be Banach algebras. Then the map $\phi:(A\widehat\otimes A) \oplus_\infty (B\widehat\otimes B) \to (A\oplus_\infty B)\widehat\otimes(A\oplus_\infty B)$ is a contractive embedding. Can ...
Albert harold's user avatar
3 votes
2 answers
324 views

projective and Haagerup tensor norms

The question below has been posted on Stackexchange few days ago but I decided to share it on MO also. Hope this is not a misuse. Fix $t\geqslant1$ and define $u_t=\pmatrix{1 & 0\\0 & t}\...
Krzysztof's user avatar
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2 answers
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How can we show that $E\:\hat\otimes_\pi\:E$ is isomorphic to a subspace of $\left(E'\:\hat\otimes_\pi\:E'\right)'$?

Let $E$ be a $\mathbb R$-Banach space $E\:\hat\otimes_\pi\:E$ denote the projective tensor product How can we show that $E\:\hat\otimes_\pi\:E$ is isomorphic to a subspace of $\left(E'\:\hat\...
0xbadf00d's user avatar
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7 votes
1 answer
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It is true that $\overline{\text{Im}(A)}\otimes \overline{\text{Im}(B)}\subset \overline{\text{Im}(A\otimes B)}$?

Let $H$ be a complex Hilbert space and $\mathcal{L}(H)$ be the algebra of all bounded linear operators on $E$. If $A,B\in \mathcal{L}(H)$, It is true that $\overline{\text{Im}(A)}\otimes \overline{\...
Student's user avatar
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1 vote
0 answers
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Non Negative Tensor Tucker Decomposition Error Degradation

I have been working on iterative decomposition methods of tensors with non negativity constraints. I have noticed that $\textbf{N}$on negative $\textbf{T}$ensor $\textbf{F}$actorization "NTF" which is ...
Mour_Ka's user avatar
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Inner Product on tensor product of Hilbert spaces is unique?

Given two Hilbert Spaces $H$ and $K$, a natural inner product on $H\otimes K$(= vector space tensor product of $H$ and $K$) is given by $\hspace{.5in}\langle h_1\otimes k_1,h_2\otimes k_2\rangle=\...
Manish Kumar's user avatar
1 vote
0 answers
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Characterizing (minimal) tensor product inside Hilbert C*-module

Let $A$, $B$ be C$^*$-algebras, $\mu$ be a state on $B$ and $\mathcal{I}$ be a family of ideals in $A$. Let $I_0:=\cap_{I\in\mathcal{I}} I$ and put $A_0:=A/I_0$. Consider the minimal tensor product on ...
This Is Me's user avatar
3 votes
0 answers
189 views

Is there a reasonable way to check intersection of these set of vectors?

Given $a,m,n,t\in\Bbb Z$, with $n=m^t$ and $a$ arbitrary, and given $\mathbb{Z}$-linearly independent vectors $v_1,\dots,v_n\in\Bbb Z^n$, and an arbitrary vector $w\in\Bbb Z^n$, such that $$\langle ...
Turbo's user avatar
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2 votes
0 answers
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Complexity of tensor decomposition vector over $\Bbb F_q$ or $\Bbb Z$

Suppose we have a matrix $$T\in\Bbb K^{n^k\times m}$$ and a target vector $v\in\Bbb F_q^m$ where $m<n^k$ and $1<k$ holds. We need to find $k$ vectors $u_1,\dots,u_k\in\Bbb K^n$ such that $$v=...
Turbo's user avatar
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3 votes
1 answer
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Tensor product of field extensions

Let $K$ be a field of characteristic 0 and $L$ a finite extension of $K$. Denote by $m$ the natural multiplication map from $L \otimes_K L$ to $L$. Denote by $I$ the kernel of the morphism $m$. Is $I$ ...
user43198's user avatar
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7 votes
1 answer
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Counting with tensor products

Suppose I've got vectors $v = (1,-1)$ and $w = (1,1)$ and any $m \in \mathbb{N}$. Let $a = v \otimes v \otimes w^{\otimes m}$ and let $\tilde{a}$ be the sum over all $\binom{m}{2}$ unique vectors ...
squiggles's user avatar
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6 votes
2 answers
709 views

Tensor product space with projective norm is incomplete

Ryan says in his book "Introduction to Tensor Products of Banach Spaces"(pg. 17) that for Banach spaces $X$ and $Y$, $X\otimes Y$ equipped with projective norm is not complete unless $X$ and $Y$ are ...
CSH's user avatar
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8 votes
2 answers
624 views

linear independent families in a tensor product

I asked this question on math.stackexchange, but without much success. Assume that $R$ is a ring (commutative, with unit) and that $M$, $N$ are two $R$-modules. Let $(e_i)_{i\in I}$ and $(f_j)_{j\in ...
Sebastian Goette's user avatar
5 votes
0 answers
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Isospectrality, Gassmann-Sunada triples, and tensor products

It is well known that Gassmann-Sunada group triples can be used to construct isospectral manifolds, arithmetically equivalent number fields, etc. (Recall that a Gassmann-Sunada triple $(U,V,W)$ ...
THC's user avatar
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8 votes
1 answer
978 views

Monoidal tensor product which preserves directed limits

Given a symmetric monoidal category $Q$, is there a construction of a (preferably full and faithful strong) monoidal embedding of $Q$ into some symmetric monoidal closed category $M$ which has all ...
Bert Lindenhovius's user avatar
7 votes
0 answers
530 views

maximal tensor product commutes with inductive limits

Let $(A_n, \phi_n)$ be an inductive system of $C^*$ algebras and let $B$ be an arbitary $C^*$ algebra. I want to prove $(\varinjlim A_n)\otimes_{max} B \cong \varinjlim (A_n \otimes_{max} B)$. This ...
Sabrina Gemsa's user avatar
3 votes
0 answers
98 views

Tensorial Construction vs Weyl Construction of Finite Dimensional Representations of GL(n, $\mathbb{C}$)?

Wu-Ki Tung discusses the "tensorial approach" to deriving all the finite dimensional irreducible representations of GL(n, $\mathbb{C}$) in chapter 13 of his book Group Theory in Physics claiming that ...
DarKnightS's user avatar
6 votes
1 answer
1k views

Tensor product of measure spaces

For a compact topological space $X$, denote by $\mathcal{M}(X)$ the Banach space of finite signed Borel (Radon) measures on $X$ with the total variation norm. This is canonically isometric to the dual ...
Matthias Ludewig's user avatar
1 vote
0 answers
96 views

Question on norms on tensor product and algebra

Question 1 Let $V,W$ be normed spaces and $\iota:V\times W \rightarrow V\otimes W$ be the canonical (algebraic) bilinear map. It can be easily shown that for any normed space $X$ and a continuous ...
Rubertos's user avatar
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Tensor Functors and Representations of Wild Quivers

A finite-dimensional $K$-algebra $A$ is of wild representation type if and only if there exists a $K\left<t_1,t_2\right>$-$A$-bimodule $_{K\left<t_1,t_2\right>}M_A$ such that the left $K\...
Iteraf's user avatar
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0 answers
207 views

An automorphism of a tensor product commuting with $\mathrm{Id}\otimes g$

Let $V$ and $W$ be vector spaces over $\mathbb{C}$, and suppose that an automorphism $f$ of $V\otimes W$ commutes with all automorphisms of the form $\mathrm{Id}_V\otimes g$, where $g \in \mathrm{Aut}(...
Y. S's user avatar
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16 votes
2 answers
433 views

exponential functors on finite dimensional complex vector spaces

Denote by $Vect^{fin}_{\mathbb{C}}$ the category of finite dimensional complex vector spaces. We will call a pair $(F,\tau)$ that consists of a functor $F \colon Vect^{fin}_{\mathbb{C}} \to Vect^{fin}...
Ulrich Pennig's user avatar
4 votes
0 answers
91 views

Simultaneous representations of elements of projective tensor products

Let $E,F$ be Banach spaces and consider the projective tensor product $E \widehat\otimes F$. If $\tau \in E \widehat\otimes F$ with $\|\tau\|<1$ then by definition we can find $(x_n)\subseteq E$ ...
Matthew Daws's user avatar
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2 votes
1 answer
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On complemented copy of $c_{0}$ in projective tensor products

Suppose that the projective tensor product of $l_{\infty}$ and $X$ contains a complemented copy of $c_{0}$. Does it follow that $X$ contains a complemented copy of $c_{0}$?
user49882's user avatar
4 votes
0 answers
173 views

Tensor product of bornological spaces and linear functionals

It is easy to see that the dual space of a bornological space $V$ (i.e. the space of bounded linear functionals) may be zero (just take any vector space with the power set as bornology). Hence in ...
Matthias Ludewig's user avatar
5 votes
0 answers
207 views

Tensors and Nuclear/Fredholm Operators

For a locally convex Hausdorff spaces $E$, consider the canonical map $$\overline{\psi}:E^\prime \hat{\otimes}_\pi E \longrightarrow L(E_\sigma)$$ that maps the projective tensor product to the space ...
Matthias Ludewig's user avatar
5 votes
1 answer
220 views

Coherent subsheaf of co-admissible modules of Schneider and Teitelbaum

Let $M$ be a co-admissible module over a Frechet Stein Algebra $A=\varprojlim A_{q_n}$ as in this paper. Let $N$ be a closed submodule of $M$. I have some difficulty in understanding lemma $3.6$ of ...
MathStudent's user avatar
1 vote
0 answers
108 views

Two tensor product norms inducing different topologies on the space of simple tensors

Are there two Normed spaces $V,W$ for which the algebraic tensor product $V\otimes W$ admits two different norms, both satisfying $\parallel x \otimes y \parallel= \parallel x \parallel. \...
Ali Taghavi's user avatar

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