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1answer
90 views

Reverse the construction of a basis for a tensor product of vector spaces [closed]

If $V,W$ are infinite-dimensional vector spaces with basis {${v_i}$} and {${w_j}$} respectively it holds that $V\otimes W$ has as basis {${v_i⊗w_j}$}. What about the reciprocal? That is: if {${v_i}$} ...
0
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0answers
15 views

Compute Frobenius inner product of two tensor-trains in terms of tensor contractions

Let $p\in\mathbb N$, $n\in\mathbb N^p$ and identify the Hilbert space tensor $\bigotimes_{k=1}^p\mathbb R^{n_k}$ with $\mathbb R^{n_1\times\cdots\times n_p}$ (equipped with the Euclidean inner product)...
20
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1answer
453 views

Semisimplicity for tensor products of representations of finite groups

Let $G$ be a group and $k$ a field of characteristic $p>0$. Let $$\rho_i: G\to GL(V_i),~ i=1,2$$ be two finite-dimensional semisimple $k$-representations of $G$, with $\dim(V_1)+\dim(V_2)<p+2.$ ...
3
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1answer
330 views

Varieties with everywhere good reduction that are isomorphic over every completion have isomorphic generic fibers

Let $R$ be the ring of integers in a number field. Let $X$ and $Y$ be smooth and proper schemes over $R$. For a maximal ideal $\mathfrak{m}\subset R$ denote the completion of the localization at $\...
3
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1answer
86 views

Generalized tensor-train decomposition

If $U\in\bigotimes_{k=1}^p\mathbb R^{n_k}$ and $U^{(k)}\in\mathbb R^{r_{k-1}}\otimes\mathbb R^{n_k}\otimes\mathbb R^{r_k}$ with$^1$ $$U_{i_1,\:\ldots\:,i_p}=\sum_{j_0=1}^{r_0}\cdots\sum_{j_p=1}^{r_p}U^...
0
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0answers
39 views

Tensor contraction (vector-valued trace) on $\bigotimes_{i=1}^k\mathcal L(E_{i-1},E_i)$

If $E_i$ is a $\mathbb R$-vector space, then the vector-valued trace $\operatorname{tr}_{E_1}:(E_2\otimes E_1^\ast)\otimes(E_1\otimes E_0)\to E_1\otimes E_0$ (or tensor contraction) is the ...
1
vote
0answers
14 views

Show that a tensor-train is contained in a recursive sequence of subspaces

Let $p\in\mathbb N$; $n_k\in\mathbb N$ and $\left(e^{(k)}_1,\ldots,e^{(k)}_{n_k}\right)$ denote the standard basis of $\mathbb R^{n_k}$ for $k\in\{1,\ldots,p\}$; $u\in\bigotimes_{k=1}^p\mathbb R^{n_k}...
0
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0answers
30 views

How are the vector-valued trace and the unique linearization of $\mathfrak L(X,Y)\:\hat\otimes_π\:X→Y$ of $\mathfrak L(X,Y)×X→Y,\;(L,x)↦Lx$ related?

Let $X$ be a $\mathbb R$-Banach space and $X'\:\hat\otimes_\pi\:X$ denote the completion of the tensor product of $X'$ and $X$ with respect to the projective norm. The trace functional $\operatorname{...
0
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1answer
45 views

Cores in the tensor-train decomposition

Let $d_i\in\mathbb N$, $I_i:=\{1,\ldots,d_i\}$ and $u\in\mathbb R^{d_1}\otimes\mathbb R^{d_2}\otimes\mathbb R^{d_3}$. It's somehow clear to me that we may regard $u$ as a three-dimensional array (see ...
0
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1answer
65 views

Determine the singular values of a compact operator in terms of the eigenvalues of an alternating tensor product of operators

Let $H$ be a $\mathbb R$-Hilbert space, $A\in\mathfrak L(H)$ be compact and $$|A|:=\sqrt{A^\ast A}$$ denote the square-root of $A$. By definition, the $k$th largest singular value $\sigma_k(A)$ of $A$ ...
8
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2answers
626 views

Is the tensor product of chain complexes a Day convolution?

Recently, Jade Master asked whether the tensor product of chain complexes could be viewed as a special case of Day convolution. Noting that chain complexes may be viewed as $\mathsf{Ab}$-functors from ...
0
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0answers
48 views

Does “tensoring” with a fixed field preserve Galois extensions of finite fields?

Let $K$ be a (possibly infinite) field of characteristic $p$, and $L$ be a finite field extension of $\mathbb{F}_p$, so that $L$ is finite and $L/\mathbb{F}_p$ is Galois. Suppose $K \otimes_{\mathbb{F}...
6
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1answer
121 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.
6
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1answer
63 views

Containment of $c_0$ in projective tensor products

Let $X$ and $Y$ be Banach spaces and denote by $X\hat{\otimes}_\pi Y$ the projective tensor product. Question: If $X\hat{\otimes}_\pi Y$ contains an isomorphic copy of $c_0$, must then $X$ or $Y$ ...
6
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1answer
106 views

Density and the projective tensor product

Let $X$ be a locally convex space (over $\mathbb{R}$), $D\subset X$ be dense, $B$ be a Banach space (again over $\mathbb{R}$) with Schauder basis $\{b_i\}_{i =1}^{\infty}$. Is the set $$ D^+\...
1
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1answer
156 views

Result of continuum tensor product of Hilbert spaces

Let's suppose that with number $\mu_1 \in \mathbb{R}$ we associate a Hilbert space $\mathcal{H}_{\mu_1}$ with countable basis $|1\rangle _{\mu_1}$, $|2\rangle _{\mu_1}$, $|3\rangle _{\mu_1}$, $\ldots$ ...
12
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3answers
1k views

Linear algebra underlying quantum entanglement?

Hope this question is appropriate. I think I saw certain claims that quantum entanglement is a certain phenomena that can be explained (or modelled) in terms of tensor products in linear algebra. I ...
2
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0answers
175 views

Tensor product of fields 2

Let $K_1, K_2$ be finite field extensions of a field $k$. Question: Is it true that $A=K_1 \otimes_k K_2$ is isomorphic to a product of group algebras over fields? Question 2: In case the ...
5
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1answer
124 views

$k$-linear $\infty$ stable categories and dg categories

This question is related to this question, where I asked about the relation between the derived category of a fiber product $Y \times_Z W$ and the push out of the diagram of derived categories one ...
2
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0answers
105 views

To understand the description of relative group homology $H_{*}(G,H;\mathbb{Z})$ in terms of free $G$-resolution

Let $G$ be a group and $H$ its subgroup ($H$ need not to be normal). Consider a chain complex $(C_{*}(G), \partial)$ where $C_n(G)$ is the free abelian group generated over the set $G^{n+1}$ and $\...
5
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1answer
167 views

Do 1-additive maps admit tensor products?

Let $\mathcal{F}$ be a set algebra (or a Boolean algebra). Following Kalton, let me call a function $f\colon \mathcal{F}\to \mathbb R$ $\delta$-additive ($\delta \geqslant 0$), whenever $f(\varnothing)...
4
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1answer
184 views

Is the tensor product of compactly generated Hausdorff abelian groups again Hausdorff?

Consider the tensor product $G \otimes_{\mathbb{Z}} H$ of two abelian groups $G$ and $H$. If $G$ and $H$ are topological groups, we can give $G \otimes_{\mathbb{Z}} H$ a topology as follows. For any $...
9
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1answer
361 views

Completed tensor product is exact

In the beginning of the 7th chapter of the book "Spectral theory and analytic geometry over non-Archimedean fields" by Vladimir Berkovich one can find the phrase "...tensor product functor is exact on ...
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0answers
86 views

Exterior paring of groups

The non-abelian tensor square $G\otimes G$ of a group $G$ is defined as a group generated by the words $g\otimes h$, $g,h \in G$ related to the conditions $gg'\otimes h=(^gg'\otimes ^gh)(g\otimes h)$...
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2answers
232 views

A description of the kernel of projection map from the tensor algebra to the symmetric algebra $T(V)\to S(V)$

Let $V$ be a vector space over some arbitrary field. Let $T(V)$ and $S(V)$ be the tensor and symmetric algebras over $V$. We have the projection map $T(V)\to S(V)$, given by $x_1\otimes\cdots\otimes ...
1
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0answers
96 views

Algebra structure on Haagerup tensor product of operator spaces

Let $A$ and $B$ be operator spaces. Is there any algebra structure on Haagerup tensor product of operator spaces such that the Haagerup tensor product becomes Banach Algebra? Any references or ideas?
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91 views

homomorphisms into tensor product algebras

Given a decomposition $H=H_1\otimes H_2$ of a Hilbert space $H$ into the tensor product of the Hilbert spaces $H_1$ and $H_2$ and a *-isomorphism $U: B(H_0)\to B(H)$, where $H_0$ is another Hilbert ...
3
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0answers
81 views

Asymmetric universal property for tensor products

Cartesian products have the following asymmetric description: $X\times Y=\coprod_{x\in X} Y$. I am looking for a similar description of tensor products (of abelian groups, for example). If $A$ is an ...
9
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1answer
329 views

Characterising natural transformations between tensor functors

I would like to know if the following conjecture is correct and if so what's a good citation for its proof. Let $\mathsf{E}$ be the category of euclidean vector spaces, i.e. objects are finite-...
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0answers
45 views

Tensor product of preordered rings

All rings in this post are commutative, unital, and contain $\frac{1}{2}$. To study "real" properties of a ring $R$, one is often interested in the orderings which exist on fraction fields of ...
2
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0answers
46 views

Rank-1 decomposability of symmetric tensors

My question is about rank-1 decomposability of symmetric tensors over the reals. Let $v_1,\dots,v_n\in\mathbb{R}^d$ be vectors. Construct the object: $$ V=\sum_{j=1}^n \underbrace{v_j\otimes v_j\...
7
votes
2answers
193 views

Do the operators in $B(E,F)$ separate points on the projective tensor product $F' \mathop{\tilde\otimes_\pi} E$?

Let $E$ and $F$ be Banach spaces, and let $\mathfrak L_{co}(E,F)$ denote the space of bounded linear operators $E \to F$ equipped with the topology of uniform convergence on the absolutely convex ...
1
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0answers
36 views

On symmetric tensors with same rank, different orders

Let $A,B$ be two symmetric tensors of same rank $m$; and orders $k$ and $\ell$, respectively. In particular, assume that $A,B$ admits the following structure: There exists $v_1,\dots,v_m\in\mathbb{R}^...
4
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1answer
219 views

Symmetric tensor decomposition

Let $T$ be an order-$k$, rank-$m$ symmetric tensor, that is, $T=\sum_{j=1}^m v_j\otimes v_j \otimes \cdots \otimes v_j$, where the Segre outer product is taken $k$ times, with $v_j\in\mathbb{R}^d$ for ...
3
votes
1answer
297 views

A submodule of a tensor product of $U_q^{\prime}(\mathfrak{g})$-modules

Does anyone have a proof for the following Lemma? Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$ and $U_q^{\prime}(\mathfrak{g})$ be the quantum affine algebra over $\...
5
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1answer
201 views

Computing a cone in a $\otimes$-triangulated category

I have a $\otimes$-triangulated category $\mathcal T$ and two triangles in $\mathcal T$: $$ x_0\to x_1\to c_x\to \Sigma x_0\ \ \ \text{and}\ \ \ y_0\to y_1\to c_y\to \Sigma y_0. $$ Consider the ...
1
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0answers
39 views

Sobolev tensor spaces and finite ranks

Let $W^{2,2}(\Omega_i)$, $\Omega_i = [-1,1]$, $i = 1,\ldots,d$ be the Sobolev spaces of twice weakly differentiable, square integrable functions. Let further $\otimes_a$ denote the algebraic tensor ...
0
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1answer
193 views

Sketching Frobenius norm of a tensor with a rank-1 random tensor

Let $A\in\mathbb{R}^{n^k}$ be a $k$-dimensional tensor with $n$ elements along each dimension. Moreover suppose $u_1,u_2,\dots,u_k\sim\text{Unif}(\pm1)^n$ are $n$ dimensional vectors with each of ...
4
votes
1answer
105 views

Linear maps preserved by algebra automorphisms

Let $A$ be a finite dimensional , associative, unital $F$-algebra, where $F$ is a field. Let $s_A:A\to F$ be an $F$-linear map. Now consider an arbitrary field extension $K/F$, and define $s_{A\...
1
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0answers
175 views

An explicit formula for characteristic polynomial of matrix tensor product [closed]

Consider two polynomials P and Q and their companion matrices. It seems that char polynomial of tensor product of said matrices would be a polynomial with roots that are all possible pairs product of ...
4
votes
1answer
332 views

Functors on the category of abelian groups which satisfy $F(G\times H) \cong F(G)\otimes_{\mathbb{Z}} F(H)$

Edit: According to the comment of Todd Trimble, I revise the question. What are some examples of functors $F$ on the category of Abelian groups or category of rings which satisfy $$F(G\times H)\cong ...
5
votes
2answers
466 views

Zero tensor product over a complex algebra?

Let $A$ be an algebra over $\mathbb{C}$. Let $M$ be a left $A$-module, let $N$ be a right $A$-module and consider the tensor product $N \otimes_A M$, which is a complex vector space. Q1: Can this ...
3
votes
1answer
160 views

Representation of $4\times4$ matrices in the form of $\sum B_i\otimes C_i$

Every matrix $A\in M_4(\mathbb{R})$ can be represented in the form of $$A=\sum_{i=1}^{n(A)} B_i\otimes C_i$$ for $B_i,C_i\in M_2(\mathbb{R})$. What is the least uniform upper bound $M$ for such $n(A)$...
3
votes
1answer
84 views

The “semi-symmetric” algebra of a vector space

If $V$ is a vector space over a field $K$, then the symmetric algebra $S(V)$ is defined as the tensor algebra $T(V)$ factorized by the two-sided ideal generated by $x\otimes y-y\otimes x$, with $x,y\...
2
votes
0answers
46 views

What is the suitable tensor product for Holder spaces

We know that for $X\subset\mathbb R^m,Y\subset\mathbb R^n$ open, then $C^0(\bar X\times\bar Y)=C^0(\bar X)\hat\otimes_\varepsilon C^0(\bar Y)$ where $V\hat\otimes_\varepsilon W$ is the injective ...
4
votes
3answers
436 views

Ideal structure of a tensor product of certain algebras

I would be grateful if anyone could give me a reference regarding the following question. Suppose that $A$ and $B$ are two unital prime algebras over a field $F$ whose center consists of scalar ...
18
votes
1answer
286 views

Interpretations for higher Tor functors

Let's work in the category $R$-${\sf mod}$, for concreteness. I know that one can see the modules ${\rm Ext}^n_R(M,N)$ as modules of equivalence classes of $n$-extensions of $M$ by $N$ (Yoneda ...
2
votes
1answer
123 views

Highest-$\ell$-weight tensor products and diagram subalgebras

Let $U_q(\mathcal{L}({\mathfrak{g}}))$ be a quantum loop algebra and $I$ the set of indexes of Dynking diagram of $\mathfrak{g}$. Consider $J\subset I$ a connected subdiagram, so that $U_q(\mathcal{L}(...
2
votes
1answer
143 views

Simple modules for direct sum of simple Lie algebras

I think that the following statement is true, but I do not know how to prove it. Let $\mathfrak{g}_1$ and $\mathfrak{g}_2$ be two real simple Lie algebras. If $M$ is a (infinite dimensional) complex ...
2
votes
1answer
225 views

On diagonal part of tensor product of $C^*$-algebras

Suppose we have a $C^*$-algebra $\mathcal{U}$, Consider the $C^*$-subalgebra generated by elements of the form $a\otimes a$, what is it isomorphic to? Is it isomorphic to $\mathcal{U}$ itself?

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