Questions tagged [tensor-products]
The tensor-products tag has no usage guidance.
356
questions
1
vote
0
answers
42
views
Is $\rho(X_1\dots X_r)^{2/r}\leq \frac{d}{r}\cdot\rho(X_1\otimes X_1+\dots+X_r\otimes X_r)$ for $d\times d$-real matries $X_1,\dots,X_r$?
Let $\rho(A)$ denote the spectral radius of a square matrix $A$. Let $r,d$ be positive integers. Let $X_1,\dots,X_r$ be $d\times d$-real matrices. Then do we necessarily have $$\rho(X_1\dots X_r)^{2/r}...
- 22.5k
2
votes
0
answers
118
views
Eigenvalues of the sum of matrices, where matrices are tensor products of Pauli matrices
recently I've been studying the toric code (a squared lattice in the context of quantum computation). I want to calculate the energy of the ground state and of all the excitations, with the respective ...
- 21
7
votes
1
answer
89
views
Projective tensor product of injective operators
I've seen claims that it is known that for a pair of bounded injective linear operators $T\colon X\to Y, S\colon W\to V$, their tensor product $T\otimes S\colon X \otimes_\pi W\to Y \otimes_\pi V$ ...
- 10.9k
2
votes
0
answers
48
views
Can the supremum of this quotient of spectral radii be reached?
Let $V$ be a finite dimensional complex inner product space. If $A_1,\dots,A_r\in L(V)$, then define a mapping $\Phi(A_1,\dots,A_r):L(V)\rightarrow L(V)$ by letting $\Phi(A_1,\dots,A_r)(X)=A_1XA_1^*+\...
- 22.5k
1
vote
0
answers
79
views
When does the Cauchy-Schwarz inequality for spectral radii of tensor products become equality?
Let $V$ be a complex finite dimensional inner product space. If $A_{1},\dots,A_{n}:V\rightarrow V$ are linear operators, then let $\Phi(A_{1},\dots,A_{n}):L(V)\rightarrow L(V)$ be the superoperator ...
- 22.5k
1
vote
1
answer
114
views
The cap set tensor in Lovett (2019)
I hope this is appropriate for the site. I am reading the paper "The analytic rank of a tensor" [S. Lovett, Discrete Analysis (2019), #7, 10 pp.] and am a bit confused in one of the ...
- 551
-1
votes
1
answer
214
views
Injection vs. bijection between $V^\star \otimes V^\star$ and $(V \otimes V)^\star$ [closed]
It is a well-known fact that, if $V$ is a vector space over a field $k$, then $V^\star \otimes V^\star$ embeds into $(V \otimes V)^\star$.
It turns out to be an isomorphism when $V$ is a finite-...
- 97
3
votes
0
answers
76
views
An example where the non-Archimedean tensor product of normed modules is only seminormed?
Let $R$ be a commutative unital ring and let $M$ be a unital $R$-module. A non-Archimedean ring seminorm on $R$ is a map $|\cdot| \colon R \rightarrow \mathbb{R}_{\geq 0}$ which satisfies
$$ | 0_R| = ...
- 31
6
votes
1
answer
308
views
Tangent bundle of a tensor product bundle
This question was also asked here on math-stackexchange.
Let $E\to M$ and $F\to M$ be vector bundles. The structure of their tangents $TE$ and $TF$ is well known. In particular, connectors map $K_E: ...
- 493
3
votes
0
answers
56
views
Dual space of Carleman functions
Let $X$ be the space of all weakly measurable functions $\gamma:\mathbb{R}^n \to L^2(\mathbb{R}^n)$ (modulo functions that are 0 almost everywhere) for which
$$\|\gamma\|_X^2 := \sup_{\|g\|_{L^2}=1} \...
- 131
3
votes
1
answer
74
views
Better solution for an evaluation over a fully connected, symmetric tensor network graph?
I have a somewhat nice approach to the following symmetric network evaluation problem. However, since the result is a possibly elaborate combinatorial problem, that I can not yet solve (edit: which ...
- 320
7
votes
2
answers
412
views
Understanding Balmer spectra
$\DeclareMathOperator\Spec{Spec}\newcommand{\perf}{\mathrm{perf}}\DeclareMathOperator\SHC{SHC}$I have just finished reading the paper "The spectrum of prime ideals in tensor triangulated ...
- 665
1
vote
0
answers
62
views
Principal component analysis with boundedness constraints
Let $A$ be an $m\times n$ matrix with entries in $F$ ($F=\mathbb{R}$ or $F=\mathbb{C}$).
It is well-known that $A$ has decompositions of the form
$$\displaystyle A = \sum_{k=1}^r\lambda_k\hspace{2mm} ...
- 1,197
9
votes
3
answers
419
views
Defining the abstract tensor product of W*-algebras via a universal property
I am playing around a bit with $W^*$-algebras, and I'm trying to come up with a definition for the $W^*$-algebraic tensor product. Here is my first attempt:
It is easy to show that such an object ...
- 263
2
votes
1
answer
180
views
Need reference for: $\lVert\cdot\rVert_{\text{max}} \leq \lVert\cdot\rVert_h$
Let $A$ and $B$ denote $C^{\ast}$-algebras. Let $\lVert\cdot\rVert_h$ and $\lVert\cdot\rVert_{\text {max}}$ denote the Haagerup norm and max $C^*$-norms on $ A \otimes B$, respectively. I am looking ...
- 971
6
votes
1
answer
185
views
What conditions are sufficient for the Leray-Hirsch theorem to be a Künneth formula?
This was originally posted on MSE, and since it didn't receive much attention, I'll try here. Let me know if this is not the appropriate place.
Given a fiber bundle $F \to E \to B$ over a paracompact ...
- 1,247
5
votes
0
answers
103
views
A concrete description of the projective tensor product of Lipschitz spaces
$\newcommand{\projtenprod}[2]{#1 \; \hat\otimes_\pi #2}$
$\DeclareMathOperator\Lip{Lip}\DeclareMathOperator\AE{AE}$
$\newcommand{\norm}[1]{\| #1\|}$
$\newcommand{\abs}[1]{| #1|}$
Background
...
- 387
10
votes
1
answer
749
views
Are condensed vector spaces over finite fields always solid?
The Clausen-Scholze theory of condensed mathematics offers an abelian category with enough projective objects that embraces the study of arbitrary locally compact (and Hausdorff) groups. The behaviour ...
- 567
3
votes
1
answer
209
views
Proof of $V\cong \overline{K} \otimes_{K} V_K$ using $H^1(G_{\overline{K}/K},\operatorname{GL}_n(K))＝0$
This is from Silverman's book "The arithmetic of elliptic curves" (AEC), p.36, lemma 5.8.1.
Lemma 5.8.1 states
Let $V$ be a $\overline{K}$-vector space, and assume that
$G_{\overline{K}/K}$ ...
3
votes
1
answer
148
views
If $A\hat\otimes B$ has identity then so are $A$ and $B$
Let $A$ and $B$ be commutative Banach algebra. I have proven that if $A$ and $B$ have identity $e_A$ and $e_B$ respectivly , then $e_A\hat\otimes e_B$ is identity for $A\hat\otimes B$ (the ...
- 763
-4
votes
1
answer
154
views
Does Rankin-Selberg convolution preserve primitivity?
Call $L$-function any element of an L-rig (see Are there infinitely many L-rigs? for a definition). Suppose $F$ and $G$ are two primitive L-functions. Is $F\otimes G$ itself primitive? If yes, does ...
- 6,672
3
votes
1
answer
171
views
Woronowicz characters are multiplicative
I'm reading the book "Compact quantum groups and their representation categories" by Neshveyev-Tuset.
Let $G$ be a $C^*$-algebraic compact quantum group with function algebra $(C(G), \Delta)...
- 263
4
votes
0
answers
135
views
Tensor product of representations on a compact quantum group
Let $\mathbb{G}$ be a $C^*$-algebraic compact quantum group (in the sense of Woronowicz) with function algebra $(C(\mathbb{G}), \Delta)$.
Let $X \in M(B_0(H)\otimes C(\mathbb{G}))$ and $Y \in M(B_0(K)\...
- 263
4
votes
2
answers
177
views
"Partially strict" monoidal categories
Recall that a monoidal category $\newcommand{\C}{\mathcal{C}} (\C,\otimes,I)$ comes equipped with an associator $\alpha_{XYZ} \colon X \otimes (Y \otimes Z) \xrightarrow{\sim} (X \otimes Y) \otimes Z$ ...
- 4,772
3
votes
0
answers
103
views
Tensor product of operator subalgebras and properties of the trace
Note that this question was already posted on MSE: https://math.stackexchange.com/questions/4290741/tensor-product-of-operator-subalgebras-and-properties-of-the-trace
Let $V$ be a vector space and let ...
- 139
6
votes
0
answers
101
views
Cech cohomology on product covers & Fréchet sheaves
My question is about the paper [Ka67].
Let $S, T$ be sheaves of nuclear Fréchet spaces over paracompact topological spaces $X, Y$, respectively; in particular, if $V \subset U$ are open subsets in $X$,...
5
votes
0
answers
56
views
Reference request: Étale base change of differential-graded algebras
I've asked this question on Math.StackExchange, but didn't receive an answer, so I'd like to try my luck here.
I'm looking for a reference for the following fact, which I've recently stumbled upon:
...
- 171
3
votes
3
answers
316
views
On the map $\Phi_M: M\otimes_RM^*\xrightarrow{x\otimes y\mapsto \left\{f\mapsto f(x)y\right\}}\text{Hom}_R(M^*,M^*) $
$\DeclareMathOperator\Hom{Hom}$Let $M$ be a finitely generated module over a Noetherian local ring $(R,\mathfrak m)$. Denote $(\_)^*:=\Hom_R(\_,R)$. There is a natural map
\begin{align} \Phi_M: M \...
- 135
0
votes
0
answers
50
views
Tensor products and intersections of modules
Is it true that $A\otimes_{\Bbbk} B \cap B \otimes_{\Bbbk} A = B\otimes_{\Bbbk} B$ if $B \subset A$ are $\Bbbk$-modules over a ring $\Bbbk$?.
It works for $\Bbbk$ a field.
Does it work in any other ...
- 1
10
votes
1
answer
302
views
Taking the category of sheaves is symmetric monoidal
Let $M$ and $N$ be topological spaces.
Let $\operatorname{Sh}(M)$ denote the presentable $\infty$-category of space-valued sheaves on $M$.
It seems to me that the equivalence
$$\operatorname{Sh}(M) \...
- 475
2
votes
1
answer
110
views
Operation including tensor product or Kronecker product transforming matrix $A$ into matrix $B$
Given two matrices $A$ and $B$:
What transformation needs to be applied to transform matrix $A$ into matrix $B$?
...
- 125
0
votes
0
answers
95
views
Operator norm on tensor product of trace classes is multiplicative
Given Hilbert spaces $\mathcal H_1,\mathcal H_2,\mathcal K_1,\mathcal K_2$ and bounded linear maps $S_i:\mathcal B^1(\mathcal H_i)\to\mathcal B^1(\mathcal K_i)$, $i=1,2$ between the respective trace ...
1
vote
0
answers
46
views
Adjacency matrix/tensor operations for graph sequences?
Consider a graph $G=(V,E)$. Its adjacency matrix $A$ is defined by $A_{u,v} = 1$ if $(u,v)\in E$, $0$ otherwise.
Consider a vector $x$ that associates a value $x_v$ to each vertex of $G$. Consider the ...
- 1,255
2
votes
1
answer
162
views
Good prime ideals in tensor products of local rings
Let $L/K$ be a field extension.
Let $R,S$ be two local commutative $K$-algebras and let $\varphi : R \to S$ be a homomorphism of $K$-algebras, not assumed to be local. Let's call a prime ideal $\...
- 56.9k
5
votes
1
answer
281
views
Tensor product of perverse sheaves on flag varieties
I am interested in computing tensor products of perverse sheaves on (partial) flag varieties. For a specific example - consider the product of the big projective on $\mathbb{P}^1$ with itself (This is ...
- 670
3
votes
0
answers
123
views
What is this SVD (called) with a singular value vector and U and V are tensors?
I am looking for information on a specific type of tensor/matrix decomposition which is quite similar to the SVD for matrices but does not look like the HOSVD since the core tensor is only a vector. ...
- 31
5
votes
1
answer
263
views
A tensor category need not be isomorphic to a strict tensor category
This question was originally posted on MSE, but got no answer even after putting a bounty on it, so I'll try here.
I'm reading the book "Tensor categories" by Etingof, Gelaki, Nikshych, and ...
- 263
3
votes
0
answers
141
views
Tensor product of associated vector bundles
Let $(P, X, \pi, G)$ and $(P', X, \pi', G')$ be two principal bundles (with Lie groups $G$, $G'$ respectively). Given a vector space $V$ and representations $\rho, \rho'$ of the Lie groups in this ...
- 181
1
vote
1
answer
98
views
The mapping $\iota \otimes S$ on the multiplier algebra
Let $A$ be a non-degenerate algebra with multiplier algebra $M(A)$. Let $S: A \to M(A)$ be an antimultiplicative linear map, i.e.
$$S(ab) = S(b)S(a).$$
Consider the mapping
$$\iota \otimes S: A \...
- 263
1
vote
1
answer
218
views
Generalization of Sinkhorn’s theorem to stochastic tensors?
Is there an algorithm for mapping a nonnegative tensor / kD array into stochastic form? i.e. assume we have a tensor of unnormalized counts, $c: ℕ^{n^k}$, can we derive a function $f: ℕ^{n^k} → ℝ^{n^k}...
- 113
4
votes
1
answer
176
views
Tensor product of positive linear maps is positive
Let $\pi_1: A_1 \to B_1$ and $\pi_2: A_2 \to B_2$ be positive linear maps between complex $*$-algebras. Is the mapping
$$\pi_1 \otimes \pi_2: A_1 \otimes A_2 \to B_1 \otimes B_2$$
again positive?
I.e.,...
- 263
7
votes
0
answers
121
views
Is this "semi-tensor product" something recently invented? Are there other usages of it?
The context: I was reading a paper in which they used the following definition called "Semi-Tensor Product" (STP) or "Cheng" product (In honor to its "inventor" D. Cheng):...
- 275
7
votes
0
answers
258
views
Generalization of a standard algebraic group theory result for a tensor problem
$\DeclareMathOperator\GL{GL}$Let $X$, $Y$, $Z$ be $\mathbb{C}$-vector spaces, and let $A\subseteq X$ and $B\subseteq Y$ and $C\subseteq Z$ be linear subspaces. Let $V=X \otimes Y \otimes Z$, acted on ...
- 874
4
votes
0
answers
108
views
Tensors of minimal rank in Schur modules $S_{\lambda}V \subset V^{\otimes |\lambda|}$
It is well known that for a vector space $V$ with $\dim(V)=n+1$ the $GL(V)-$module $V^{\otimes d}$ splits as a sum of irreducible representations (with suitable multiplicities) $S_{\lambda}V$, where $\...
- 1,167
3
votes
3
answers
259
views
Universal property of tensor products of bounded operators
Consider the tensor product of bounded operators. Does this tensor product satisfy the universal property of the tensor product, i.e., for any bilinear map $F: B(\mathcal H_1)\times B(\mathcal H_2)\to ...
- 750
3
votes
0
answers
123
views
Spectral norm and "operator norm" for hypergraphs
Consider a $d$-regular, $k$-uniform hypergraph: the elements $S$ of its set $E$ of edges are subsets of $V$ of size $k$, and each vertex $v\in V$ is in $d$ edges. We can then define its adjacency ...
- 16.7k
10
votes
1
answer
403
views
Basis of invariant tensors of rank n in three dimensions
[This is a question motivated by theoretical physics, so apologies if the language is rough...]
In three dimensions the spaces of invariant (or isotropic) tensors of rank $n$ have dimensions 1, 0, 1, ...
- 1,018
2
votes
1
answer
103
views
Inclusion $M(A) \otimes M(B)\subseteq M(A\otimes B)$ of multiplier algebras
Consider the following definitions given in Timmerman's book "An invitation to quantum groups and duality":
m
Further in the book, it is claimed that if $A$ and $B$ are non-degenerate ...
user167952
6
votes
1
answer
492
views
If $M\otimes_S T$ is an $A$-module, is $M$ an $A$-module?
Let $\mathbb{C}$ be the field of the complex numbers. Let $R=\mathbb{C}[x]$, $T=\mathbb{C}\langle x\rangle$ be the ring of entire series with convergence radius at least $1$, and let $S=\mathbb{C}\...
- 1,243
0
votes
0
answers
119
views
Definition of tensor product of dense subspaces of Hilbert spaces
Let $\mathscr{H}_{1}$ and $\mathscr{H}_{2}$ be Hilbert spaces. If $\psi_{1}\in \mathscr{H}_{1}$ and $\psi_{2}\in \mathscr{H}_{2}$, define $\psi_{1}\otimes \psi_{2}$ to be a function on $\mathscr{H}_{1}...
- 1,027