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Can we test if an abelian group is finitely generated by taking tensor product?

If $A$ is a finitely generated abelian group, then we know that for all fields $k$, the tensor product $A\otimes_{\mathbb{Z}}k$ is finite dimensional as a $k$-vector space. The converse is not true, ...
Bingyu Zhang's user avatar
3 votes
0 answers
197 views

Stokes' lemma / Yoneda's theorem

Definition: Write SmoothOrientedManifold for the category of smooth oriented real manifolds. Definition: Write ChainComplex for the category of chain complexes, and write $\otimes$ for the tensor ...
Ron Z.'s user avatar
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8 votes
2 answers
380 views

Irreducible tensor product representations in finite simple groups

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\PSp{PSp}\DeclareMathOperator\PSU{PSU}$Background: A representation $ \rho: G \to \GL(V) $ of a group $ G $ on a (complex) ...
Sebastien Palcoux's user avatar
2 votes
0 answers
129 views

Tensor product of finite extensions of $\mathbb{Q}_p$

Consider the tensor product of finite extensions of a field $F$ of characteristic zero. (I am interested in the case $F=\mathbb{Q}_p$.) $(1)$ If $M$ is a finite Galois extension of $F$ with Galois ...
ZZP's user avatar
  • 590
1 vote
0 answers
142 views

Vanishing (infinite) tensor products

Since the advent of free probabilities and QFT, infinite tensor products of $R$-associative algebras with units has become more familiar to the working mathematician. Starting from the (permuting) ...
Duchamp Gérard H. E.'s user avatar
2 votes
1 answer
197 views

Combination of simple tensors - II

This is a follow-up question to Combination of simple tensors. I am interested in devising an alternative norm (I mean, other than the usual $\pi$ or $\epsilon$ norms) in the tensor product of two ...
Lorenzo Guglielmi's user avatar
3 votes
0 answers
147 views

Multiplicative structure on Čech–Alexander complexes

I have the following basic question on Čech–Alexander complexes. Let $R$ be a ring and $A$ be an $R$-algebra. To this datum one can attach a cosimplicial ring which assigns to an object $[n]=\{0,1,\...
Stabilo's user avatar
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1 vote
1 answer
115 views

Carnot–Carathéodory norm and the inner product norm

It is well-known that given the extended tensor algebra $T((\mathbb{R}^d))$ one may extract a separable Hilbert space by considering the subset $$T^1((\mathbb{R}^d)) := \left\{h \in T((\mathbb{R}^d)) :...
Gaspar's user avatar
  • 91
2 votes
1 answer
221 views

Combination of simple tensors

I aksed this question on Math Stack Exchange 6 days ago, with no answer: https://math.stackexchange.com/q/4875445/1297919 Let $X$ and $Y$ two Banach spaces and let $X\otimes Y$ their tensor product. ...
Lorenzo Guglielmi's user avatar
3 votes
1 answer
129 views

Is there any (specially Algebraic Geometrical) exposition of Koike Terada's Young-diagrammatic methods for the representation theory paper?

I am talking about the paper by Koike, Kazuhiko and Terada, Itaru, Young-diagrammatic methods for the representation theory of the classical groups of type ($B_n$), ($C_n$), ($D_n$), J. Algebra 107, ...
Dibyendu's user avatar
1 vote
0 answers
27 views

Inner product of signatures of piecewise linear paths

It is a well-know observation that, given two points $x_1,x_2 \in \mathbb{R}^d$, the path signature associated to their linear interpolation is given by the tensor exponential. Precisely, if $\Delta x$...
Gaspar's user avatar
  • 91
2 votes
1 answer
160 views

Are projective tensor products left-exact if one considers only maps of norm at most 1?

Consider the category $\mathrm{Ban}$ of Banach spaces and bounded linear maps and the category $\mathrm{Ban}_1$ of Banach spaces and bounded linear maps of operator norm at most 1. Let $\otimes_\pi$ ...
Stephan Mescher's user avatar
4 votes
0 answers
112 views

Weakly null sequences in projective tensor products II

The question in this post is the question below from an article by Rodriguez & Rueda Zoca [1]. Below is a complimentary salad/side dish that accompanies the main course. Let $B^2(X,Y)$ denote ...
Onur Oktay's user avatar
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4 votes
0 answers
146 views

Isomorphic copies of $c_0$ in the projective tensor products

There exist Banach spaces $X$ such that the projective tensor product $X\mathbin{\hat{\otimes}}_\pi X$ contains an isomorphic copy of $c_0$ [BourgainPisier1983]. Moreover, $X$ is an $\mathcal{L}_\...
Onur Oktay's user avatar
  • 2,283
0 votes
0 answers
122 views

On a matrix equation with Kronecker product

Is there any work on the matrix equation in unknowns $X, Y \in {\Bbb C}^{n \times n}$ $$(X \otimes Y + Y \otimes X) \operatorname{vec}(A)=0$$ where $\otimes$ is the Kronecker product? Or, in general, ...
mukhujje's user avatar
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1 vote
1 answer
134 views

Diagonalizability, orthogonal diagonalizability of higher order tensors and their being or not being dense in some suitable topology

For our discussion, we'll assume that we're working with $\mathbb{R}^m$ only, but much or all of the following discussion should be carried over immediately to any finite dimensional inner product ...
Learning math's user avatar
4 votes
0 answers
125 views

Is there a fast way to do this tensor power/trace operation?

Well, I asked this question on Math SE, and didn't get any responses, so I'm trying it here. Given an $M*N*P$ tensor $T$, is there a fast way of computing the following "eighth-power trace"? ...
Craig's user avatar
  • 525
1 vote
1 answer
180 views

Tensor product of faithful normal states is faithful

I know that given C*-algebras $A, B$ with faithful states $\omega,\varpi$, the tensor product state $\omega\otimes\varpi$ on the minimal tensor product $A\otimes_{\text{min}}B$ is faithful. I also ...
J_P's user avatar
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8 votes
1 answer
594 views

A question regarding symmetrizing the tensor product of vectors in two different ways

Let $V = \mathbb{C}^m$, endowed with the standard hermitian inner product which we will denote by $\langle \cdot, \cdot \rangle$, $n$ be a positive integer and $\Sigma_n$ denote the symmetric group on ...
Malkoun's user avatar
  • 5,041
1 vote
1 answer
139 views

Do completely bounded maps on an operator space have a completely contractive Banach algebra structure?

Let $X$ be an operator space and $CB(X)$ be the set of all completely bounded linear maps $f: X \to X$. Note that $CB(X)$ becomes a Banach algebra for the composition of operators. Is the ...
Andromeda's user avatar
  • 169
2 votes
1 answer
165 views

$(S\otimes T)^{it}= S^{it}\otimes T^{it}$ 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 ...
Andromeda's user avatar
  • 169
0 votes
0 answers
68 views

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 ...
Martin Geller's user avatar
3 votes
0 answers
106 views

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^*(...
Tomás Pacheco's user avatar
3 votes
0 answers
162 views

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 ...
Mikhail Bondarko's user avatar
3 votes
1 answer
130 views

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 ...
Andromeda's user avatar
  • 169
15 votes
3 answers
2k views

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. ...
Daniel Asimov's user avatar
7 votes
2 answers
323 views

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 ...
Bence Racskó's user avatar
7 votes
2 answers
770 views

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 ...
Nanoputian's user avatar
4 votes
0 answers
233 views

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 $A$ ...
Math Lover's user avatar
  • 1,105
1 vote
0 answers
45 views

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 (...
mathieu_matheux's user avatar
1 vote
0 answers
172 views

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\...
tudong's user avatar
  • 41
1 vote
1 answer
144 views

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 ...
Sanae Kochiya's user avatar
3 votes
0 answers
184 views

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 (...
M-M's user avatar
  • 31
5 votes
0 answers
163 views

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 $A\otimes A$...
Ben's user avatar
  • 970
3 votes
0 answers
57 views

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 \...
Seven9's user avatar
  • 515
13 votes
0 answers
167 views

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 $(A,B)$...
Alcides Buss's user avatar
2 votes
0 answers
145 views

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^...
Onur Oktay's user avatar
  • 2,283
1 vote
0 answers
127 views

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) ...
Snake Eyes's user avatar
0 votes
0 answers
166 views

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\...
Onur Oktay's user avatar
  • 2,283
0 votes
1 answer
86 views

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 ...
user443060's user avatar
10 votes
1 answer
950 views

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 ...
dm82424's user avatar
  • 360
2 votes
0 answers
344 views

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 ...
Onur Oktay's user avatar
  • 2,283
1 vote
0 answers
119 views

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 $\otimes:(v, w) \mapsto v \otimes w$ ...
amilton moreira's user avatar
1 vote
0 answers
141 views

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 ...
Meisam Soleimani Malekan's user avatar
0 votes
0 answers
245 views

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 ...
prochet's user avatar
  • 3,442
0 votes
0 answers
207 views

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 $A$ ...
Dasherman's user avatar
  • 203
13 votes
2 answers
431 views

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 ...
Libli's user avatar
  • 7,240
3 votes
1 answer
305 views

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 ...
David Corwin's user avatar
  • 15.2k
1 vote
1 answer
122 views

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 \...
J. De Ro's user avatar
  • 508
2 votes
1 answer
154 views

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 ...
Grwlf's user avatar
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