# Questions tagged [tensor-products]

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### 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}...

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### 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 ...

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### 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 $T\otimes S\colon X \otimes_\pi W\to Y \otimes_\pi V$ ...

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### 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^*+\...

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### 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 ...

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### 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 ...

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### 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-...

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### 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| = ...

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### 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: ...

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### 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} \...

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### 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 ...

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### 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 ...

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### 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} ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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
...

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### 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 ...

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### 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
$G_{\overline{K}/K}$ ...

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### 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 ...

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### 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 ...

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### 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)...

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### 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)\...

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### "Partially strict" monoidal categories

Recall that a monoidal category $\newcommand{\C}{\mathcal{C}} (\C,\otimes,I)$ comes equipped with an associator $\alpha_{XYZ} \colon X \otimes (Y \otimes Z) \xrightarrow{\sim} (X \otimes Y) \otimes Z$ ...

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### 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 ...

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### 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 $X$,...

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### 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:
...

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### 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 \...

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### 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 ...

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### 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) \...

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### 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 $B$?
...

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### 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 ...

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### 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 ...

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### 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 $\...

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### 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 ...

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### 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. ...

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### 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 ...

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### 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 ...

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### 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 \...

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### 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}...

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### 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.,...

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### 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):...

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### 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 ...

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### 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 $\...

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### 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 ...

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### 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 ...

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### 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, ...

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### 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 ...

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### 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}\...

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### 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}...