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Questions tagged [tensor-products]

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6
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0answers
169 views

The symmetric power of a tensor product

In the representation theory, if $S^{\lambda}(V)$ is the irreductible representation of $\text{GL}(V)$ associated to a partition $\lambda \vdash n$ (in perticular, $S^n(V)$ is the $n^{\text{th}}$ ...
0
votes
1answer
26 views

tensor stability of block-positive matrices

Let $X_{AB}$ be an operator acting on the tensor-product Hilbert space $\mathcal{H}_A \otimes \mathcal{H}_B$. Suppose that $X_{AB}$ is block positive, meaning that (in Dirac notation) $\langle \psi |...
0
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0answers
15 views

How to derive a bound of distortion / error between two different tensor decompositions

Consider a tensor $\mathcal{X}\in\mathbb{R}^{I\times J\times K}$. It can be approximately decomposed/factored in multiple ways. Namely by using the TUCKER3 decomposition: $\mathcal{X}\approx \sum_{p=...
1
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0answers
25 views

Decomposition into irreducible of a representation of the wreath product $S_d \wr S_n$ (3)

Let: $$ R_m^n= \bigl( F^{\widetilde{\otimes n-m}} \boxtimes S^{\widetilde{\otimes m}} \bigr)\bigl\uparrow_{S_{n-m} \times S_{m}}^{S_m} : $$ This is an irreducible representation of $S_d \wr S_n$. I'd ...
4
votes
1answer
125 views

Global dimension of the tensor algebra

Let $R$ be a semisimple ring with a non-zero $R$-bimodule V. Let $T_R(V):= \bigoplus\limits_{k=0}^{\infty}{V^{\otimes_k}}$ be the tensor algebra of $V$. Question 1: Is there a simple proof that $...
1
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0answers
57 views

When are “square spans” not transversal?

Let $V$ be a finite-dimensional vector space over a field $K$. Given a basis $\{v_1,\dotsc,v_n\}$ for $V$, we define the "square span" of the basis to be the subspace of $V\otimes V$ spanned by $v_1\...
3
votes
0answers
75 views

Jacobson radical of a tensor product

Let $R$ be a commutative ring and $A_1$, $A_2$ be $R$-algebras. Is there any general mean to compute the Jacobson radical of the tensor product $A_1\otimes_R A_2$ in terms of ${\rm Rad}(A_i )$, $i=1,2$...
7
votes
2answers
248 views

How do fractional tensor products work?

[I asked and bountied this question on Math SE, where it got several upvotes and a comment suggesting it was research-level, but no answers. So I'm reposting here with slight edits, but please feel ...
6
votes
1answer
204 views

Direct proof of “Nuclear implies $C_{red}^*(G) \cong C^*(G)$”

It is well-known that for a discrete group $G$ the following statements are equivalent: $C_{red}^*(G)$ nuclear $C_{red}^*(G) \cong C^*(G)$ canonically i.e. there exists an *-isomorphism between the ...
4
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0answers
64 views

On ideals in Noetherian rings, isomorphic to the trace of some finitely generated module

Let $R$ be a Noetherian ring. For a finitely generated $R$-module $M$, let $tr_R(M):=Im(\tau_M)$, where $\tau_M:M\otimes Hom(M,R)\to R$ is the map defined as $\tau_M(m\otimes f)=f(m)$. Let $I$ be a ...
3
votes
1answer
193 views

tensor product of massless Poincare representations

Consider two massless representations of the connected Poincare group $ISO_0(1,3)$ with helicities $s$ and $t$. What is the decomposition of their tensor product into irreducibles? Massless ...
3
votes
1answer
115 views

Is the norm of the Banach space projective tensor product of finite-dimensional C*-algebras a C*-norm?

Let $A$ and $B$ be two finite-dimensional C*-algebras. Let $\gamma$ denote the projective Banach space tensor product norm on the algebraic tensor product $A\odot B$, so $\gamma(t)=\inf\{\sum_{i}\|...
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0answers
66 views

Shortest vectors in tensor product and maximal lattices in tensor product

$\mathcal L$ and $\mathcal L'$ be full rank lattices in $\mathbb R^n$ with shortest vectors $v_1,\dots,v_n$ and $v_1',\dots,v_n'$ respectively where $$\|v_1\|_2\leq\dots\leq\|v_n\|_2$$ $$\|v_1'\|_2\...
5
votes
0answers
124 views

A tensor product for dg-categories

For a finite group denote by $\mathbf{Ch}^G$ the dg-category of $G$-representations in chain complexes over a field. Is there a tensor product $\otimes$ of dg-categories (similar to the Deligne ...
1
vote
0answers
47 views

Necessary conditions for a function to be represented as symmetric tensor product

Let $f(\cdot,\cdot)$ be any function of 2 arguments. Suppose also that the following equation holds $$f(y,x)+f(x,y)=g(x) h(y) +g(y)h(x) \quad \forall x, y$$ for arbitrary $h(\cdot)$ and $g(\cdot)$. ...
0
votes
0answers
46 views

Problem of expressing a two matrices product

I think need a different method of matrix multiplication here: Suppose I have two matrices: $A = (a_{ij}) \in \mathbb{R}^{m \times m}$ and a partitioned matrix $B = (B_{ij}) \in \mathbb{R}^{mp \times ...
3
votes
1answer
170 views

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 ...
1
vote
1answer
116 views

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 $F$ ...
2
votes
1answer
125 views

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 ...
4
votes
1answer
146 views

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 ...
2
votes
1answer
100 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 ...
7
votes
1answer
166 views

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 ...
1
vote
1answer
85 views

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\...
3
votes
1answer
107 views

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}}...
7
votes
0answers
118 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 ...
1
vote
1answer
82 views

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 ...
1
vote
0answers
72 views

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 ...
1
vote
0answers
54 views

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$^1$ $$\...
1
vote
1answer
125 views

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{...
1
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0answers
65 views

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\...
2
votes
0answers
54 views

Tensor product and quotients of it

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?(...
2
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0answers
108 views

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 \...
4
votes
1answer
161 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,...
4
votes
1answer
171 views

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=\...
3
votes
1answer
124 views

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, ...
1
vote
1answer
93 views

Tensor product and hyponormality of operators

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\...
3
votes
1answer
176 views

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 ...
4
votes
1answer
115 views

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 \...
0
votes
1answer
75 views

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 ...
3
votes
2answers
221 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}\...
4
votes
2answers
172 views

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\...
6
votes
1answer
400 views

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{\...
1
vote
0answers
20 views

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 ...
3
votes
0answers
308 views

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=\...
1
vote
0answers
40 views

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 ...
-2
votes
1answer
193 views

$G \oplus \mathbb Z^k \cong G \oplus \mathbb Z^l $ but $(G \oplus \mathbb Z^m)\otimes \mathbb Q \ncong (G \oplus \mathbb Z^n)\otimes \mathbb Q$? [closed]

Does there exist non-negative integers $k,l,m,n$, with $k \ne l$, such that $G \oplus \mathbb Z^k \cong G \oplus \mathbb Z^l $ but $(G \otimes_{\mathbb Z}\mathbb Q) \oplus \mathbb Q^m \ncong (G \...
0
votes
0answers
60 views

On expected intersection of typical lattice with tensor product of cubes?

Consider $T$ tensor product of $t$ origin centered boxes $$\underbrace{[-m_i,m_i]\times\dots\times[-m_i,m_i]}_{n_i\mbox{ times}}$$ each in $\Bbb R^{n_i}$ where $i\in\{1,2,\dots,t\}$ holds. If we ...
3
votes
0answers
183 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 ...
1
vote
0answers
101 views

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=...
0
votes
1answer
395 views

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 $I$ ...