# Questions tagged [tensor-products]

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396
questions

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### $(S\otimes T)^{it}= S^{it}\otimes T^{it}$ for unbounded operators

Let $S,T$ be unbounded, closed operators in Hilbert spaces $H,K$. In that case, we can form the tensor product operator $S\otimes T$ on the Hilbert space $H\otimes K$ which is the closure of the ...

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### Continuity of linear map on tensor product spaces with different norm properties

I originally asked this question on StackExchange, but I think that it may be more suitable to here.
Let $V$ and $U$ be Banach spaces. I'm considering a linear map $\phi: V \rightarrow U$, and ...

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### Reference request for embedding of a tensor product $C^*$-algebra

I am studying Ruy Exel's paper "A new look at the crossed product of a $C^*$-algebra by a semigroup of endomorphisms." In the proof of Theorem 11.7 he writes:
Let $G$ be ameanable, thus $C^*(...

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### Which "tensor" endofunctors on triangulated categories are essentially exact?

Assume that $T$ is a symmetric monoidal triangulated category, and $X$ is an object in it. Then the functors $X\otimes -$ and $-\otimes X: T\to T$ are not necessarily exact since they send ...

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### Property that follows from conditions involving slice maps on Hilbert module

Let $A,B$ be $C^*$-algebras and $E$ be a right $A$-Hilbert $C^*$-module. We can form the Hilbert $A\otimes B$ (minimal tensor product) module $E \otimes B$. If $\omega \in B^*$, there is a unique ...

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### Tensor product of vector bundles

The Whitney sum (where fibre dimensions add) of two real, or two complex, vector bundles $\pi : E \to X$ and $\pi' : E' \to X$ over a topological space $X$ is not hard to get an intuitive grasp of. ...

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### Decomposition of tensors into symmetry classes according to Schur functors

I am mainly looking for references on this subject, as I was unable to find any, at least any that answers what I am looking for to a satisfactory degree.
As it is well-known and extremely easy to ...

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### Tensor product of irreducible representations of an algebra

Let $A$ be an associative algebra over $\mathbb{C}$ with irreducible finite-dimensional representations on $V$ and $W$. Then is the tensor product of representations on $V \otimes W$ semi-simple?
The ...

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### About injective and projective tensor products of commutative Banach algebras

Let $A_i$, $B_i$, $i=1,2$, be semisimple commutative Banach algebras such that $A_i$ is isomorphic to $B_i$, $i=1,2$. Is the injective tensor product of $A_1$ and $A_2$ is isomorphic to the injective ...

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### The ideal structure of the Haagerup tensor product of $C^{\ast}$-algebras

I'm currently reading the paper The ideal structure of the Haagerup tensor product of $C^{\ast}$-algebras and having difficulty in understanding the proof of Proposition $4.5$ from the paper.
Let $A$ ...

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### Intersection of (sub-)modules under Laurent and formal rings

I have a (hopefully quick) question regarding an intersection of two tensor modules. Let K
be a field and $A,B$ finitely-generated modules over the Laurent series $K((X))$. Let $\tilde{A}$ be a (...

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157
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### The conditions to determine whether multivector $\Lambda\in\wedge^k V$ is decomposable

In Section 5, Chapter 1 of the famous book "Principles of algebraic geometry" by Griffiths and Harris, there are two equivalent conditions to determine whether a multivector $\Lambda\in\...

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1
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128
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### Convergence of the partial sum of a sequence strictly converging to zero

The following question comes from a statement in Lemma 16.4 in K-theory and $C^{\ast}$-Algebras written by N.E. Wegge-Olsen. Let $A$ be a non-unital $C^*$-algebra, $\{p_n\}_{n\in\mathbb{N}}$ be a ...

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151
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### Intersection of two modules (and sub-modules) under tensors

I have a (hopefully quick) question regarding an intersection of two tensor modules. Let $K$ be a field and $A, B$ finitely-generated modules over the Laurent series $K((X))$. Let $\tilde{A}$ be a (...

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150
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### Maximal minors of tensor product

Let $r \leq n$ be integers, and let $A$ be an $r \times n$ integer-valued matrix such that each $r\times r$ minor of $A$ is in $\{0, 1,-1\}$. Is it true that each $r^2 \times r^2$ minor of $A\otimes A$...

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### Automatic complete boundedness for bilinear and multilinear maps

$\newcommand{\cb}{\mathrm{cb}}$Let $T : X \rightarrow Y$ be a bounded linear map between Banach spaces. We have the following results concerning automatic complete boundedness:
$\|T : X \rightarrow \...

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### Existence of more than two C*-norms on algebraic tensor product of C*-algebras

Let $A$ and $B$ be two C*-algebras. Then $(A,B)$ is called is a nuclear pair if there is a unique $C^*$-norm on the algebraic tensor product $A\odot B$.
If $A$ or $B$ is nuclear, then all pairs $(A,B)$...

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135
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### Quotients of $c_0\mathbin{\hat{\otimes}_{\pi}}\ell^1$

Let $\hat{\otimes}_{\pi}$ denote the projective tensor product. Let $$\mathcal{S} = \{V\subseteq c_0\mathbin{\hat{\otimes}_{\pi}}\ell^1\textrm{ closed subspace}: {c_0\mathbin{\hat{\otimes}_{\pi}}\ell^...

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### On Noetherianity and local ness of a completed tensor product

Let $R$ be a regular local complete (with respect to the maximal ideal) ring with field of fraction $K$. Let $S\cong R[[x_1,\cdots, x_n]]/J$ (this is a Noetherian local ring which is an $R$-algebra) ...

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### Completely continuous maps from projective tensor products into $c_0$

Let $E$, $F$ be two Banach spaces and $E\mathbin{\hat{\otimes}}_{\pi}F$ denote their projective tensor product.
For each unit norm $\xi\in E$ and $\gamma\in F$, let's define
$$
J_{\gamma}:E\to E\...

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79
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### Proving finite presentation [closed]

Let $R$ be an integral domain, $S$ be a finitely presented $R$ algebra. Then for a flat $R$ module $M$ which is also a finitely generated $S$ module I need to show that $M \otimes_{R}T$ is a fintely ...

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### Clebsch–Gordan decomposition formula for algebraic groups

$\DeclareMathOperator\SL{SL}$There is a well-known Clebsch–Gordan decomposition formula for irreducible representations of $\SL_2$. If $V_n$ denotes the unique $n+1$-dimensional irreducible ...

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329
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### Weakly null sequences in projective tensor products

First, I'd like to record a question that may still be open. The snippet below is taken from DiestelPuglisi2009.
Second, let $E$ be a Banach space, $(u_n)$ be a weakly null sequence in the projective ...

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### Does Schwartz kernel theorem come from the universal property of tensor product?

In wikipedia we have Tensor product
The tensor product of two vector spaces $V$ and $W$ is a vector space denoted as $V \otimes W$, together with a bilinear map $\otimes:(v, w) \mapsto v \otimes w$ ...

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138
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### Could we characterize elements in the second dual by the character space?

Let $A$ and $B$ be two semisimple commutative Banach algebras. Assume that $A\mathbin{\tilde\otimes} B$ is a Banach algebra obtained by completing $A\otimes B$ with respect to a cross norm not ...

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183
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### completion and tensor product

Let $A$ be a commutative ring, consider the map $Spec(A[[t]])\rightarrow Spec(A)$, does it have geometrically connected fibers?
If $A$ is noetherian, it is clear because one has for $k$ a residue ...

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131
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### Eigenvalue multiplicity of tensor product of positive operator with itself

Let $H$ be a separable complex Hilbert space and let $A\in B(H)$ be positive with $||A||=1$ and have eigenvalue 1 with multiplicity 1. Suppose $A=T^*T$ for some $T\in B(H)$. Denote the spectrum of $A$ ...

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### Tensor product of finite type UFD algebras over an algebraically closed field is again UFD?

Let $K$ be an algebraically closed field, $A$ and $B$ two finite type $K$-algebras which are assumed to be UFD. Is $A \otimes_K B$ again a UFD?
This question has been already asked here and here, but ...

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### Is Spec of a ring monoidal or anti-monoidal?

Let $A$ and $B$ be rings. A very senior mathematician impressed on me the importance of writing
$$
\operatorname{Spec}{A \otimes B} = \operatorname{Spec}{B} \times \operatorname{Spec}{A}
$$
One can ...

1
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1
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102
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### Which elements live in the image of the canonical map $X \otimes_\mathcal{F} M \to B(M_*, X)$?

Let $X\subseteq B(H)$ be an operator system and let $M\subseteq B(K)$ be a von Neumann algebra. We form the Fubini-tensor product
$$X \otimes_\mathcal{F} M := \{z \in B(H\otimes K): (\sigma\otimes \...

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### Explain how to infer a density matrix from the statistics of quantum measurements

This question follows the "Probabilistic Simulation of Quantum Circuits with the Transformer" paper by Carrasquilla et al. In the Formalism section on page 2 the authors state that ...

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### Tensor product is complete?

Let $(V,\|\cdot\|_V)$ and $(W,\|\cdot\|_W)$ be Banach spaces and let the norm $\|\cdot\|_{V\otimes W}$ on the tensor product space $V\otimes W$ be admissible in the following sense: for $v\in V, w\in ...

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### Multiplier algebra of Fock space

For any vector space, one may form the tensor algebra with multiplication being the tensor product. For a Hilbert space $\mathcal{H}$, the analogous construction is the Fock space
$$
\mathcal{F}(\...

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1
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### What resource do Markov and Shi mean when they estimate tensor contraction complexity?

Markov and Shi in their paper Simulating quantum computation by contracting tensor networks define the contraction complexity as follows (page 10):
The complexity of π is the maximum degree of a ...

4
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### Certain isotypical component of the tensor product of irreducible representations of $\mathrm{U}(n)$

The following question is closely related to this one.
Let $\mathrm{U}(n)$ be the group of all (complex) unitary matrices $n\times n$. It is known that all irreducible representations of $\mathrm{U}(n)...

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### Is the rank preserved when the spectral radius is maximized?

If $A$ is a matrix, then let $\rho(A)$ denote the spectral radius of $A$. If $A=(a_{i,j})_{i,j}$, then let $\overline{A}=(\overline{a_{i,j}})_{i,j}$.
Suppose that $A_1,\dots,A_r\in M_{n}(\mathbb{C})$ ...

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### Recoverability of categorical products in graphs

For simple, undirected graphs $G, H$, let $G\times H$ denote the categorical product, or tensor product, of $G$ and $H$.
Let us call a graph $G = (V,E)$ (product-)reducible if there are graphs $G_i = ...

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### Tensoring with an induced representation: proof question

Let $G$ be a locally compact Hausdorff group and $H$ a closed subgroup of $G$. If $\sigma: H \to B(\mathcal{K}_\sigma)$ is a unitary representation of $G$, we can associate an "induced ...

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### Tensor product of exact couples

Suppose $(D_1,E_1;i_1,j_1,k_1)$ and $(D_2,E_2;i_2,j_2,k_2)$ are two exact couples, ie., there are exact sequences $D_1\xrightarrow{i_1}D_1\xrightarrow{j_1}E_1\xrightarrow{k_1}D_1$ and $D_2\xrightarrow{...

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### An inequality for the spectral radius of block matrices

Let $d,m$ be positive integers. Suppose that $A_{i,j}$ is a $d\times d$-matrix with real entries whenever $i,j\in\{1,\dots m\}$.
Let $A$ be the $dm\times dm$ matrix that can be written as a block ...

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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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1
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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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1
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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} \...