Questions tagged [tensor-products]
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6
votes
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
votes
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
vote
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
vote
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
votes
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}\|...
1
vote
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
vote
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
votes
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 $I$ ...
