Questions tagged [tensor-products]

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Who stated and proved the "Hopf lemma" on bilinear maps?

If $A\otimes B\rightarrow C$ is a nondegenerate linear map, where $A, B, C$ are vector spaces over an algebraically closed field, then $\dim C\ge \dim A + \dim B -1$. Nondegenerate here means that ...
quim's user avatar
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13 votes
0 answers
321 views

When does Hochschild homology commute with infinite products?

Let $A$ be an associative algebra. Its zeroth Hochschild homology $\mathrm{HH}_0(A)$ is the cokernel of the linear map $A^{\wedge 2} \to A$, $a \wedge b \mapsto ab - ba$. I.e. you quotient the ...
Theo Johnson-Freyd's user avatar
13 votes
0 answers
557 views

Symmetric (extended) Haagerup tensor product

Given a von Neumann algebra M, then the weak$^*$ (or extended) Haagerup tensor product of M with itself is the collection of $\tau\in M\overline\otimes M$ with $$\tau=\sum_i x_i\otimes y_i$$ the sum ...
Matthew Daws's user avatar
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12 votes
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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 $(A,B)$...
Alcides Buss's user avatar
11 votes
0 answers
408 views

Is there a notion of tensor product of perfect bases of representations of Lie algebras?

Berenstein and Kazhdan define perfect bases as an "unquantized" version of crystal bases. A perfect basis is roughly a basis with a crystal structure such that $E_i\cdot v=\mathbb{C}\cdot \tilde{e}...
Ben Webster's user avatar
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10 votes
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505 views

Tensorial decomposition of $B(H)$

Let $H$ be an infinite-dimensional Hilbert space and let $\mathcal{B}(H)$ be the (C*/W*-)algebra of bounded operators on it. Actually, you may forget about the involution in $\mathcal{B}(H)$ because I ...
TrzyTrypy's user avatar
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9 votes
0 answers
427 views

Objects and morphisms in Kelly's tensor product of finitely cocomplete categories

Let $k$ be a field. Let $\mathcal{C},\mathcal{D}$ be finitely cocomplete $k$-linear categories, which are essentially small. Then Kelly's tensor product $\mathcal{C} \boxtimes \mathcal{D}$ is a ...
Martin Brandenburg's user avatar
8 votes
0 answers
264 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 ...
Ben's user avatar
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8 votes
0 answers
331 views

Torsion in a tensor product over a group ring

Let $\Gamma$ be a finitely generated dense subgroup of a pro-$p$ group $G$. Let $\mathbb Z_p$ be the ring of $p$-adic numbers. Denote by $\mathbb Z_p[[G]]$ the completed group algebra. Is it true ...
Andrei Jaikin's user avatar
7 votes
0 answers
530 views

maximal tensor product commutes with inductive limits

Let $(A_n, \phi_n)$ be an inductive system of $C^*$ algebras and let $B$ be an arbitary $C^*$ algebra. I want to prove $(\varinjlim A_n)\otimes_{max} B \cong \varinjlim (A_n \otimes_{max} B)$. This ...
Sabrina Gemsa's user avatar
7 votes
0 answers
263 views

Do there exist "non-algebraic tensor products" for "algebraic" triangulated categories?

Let us call a triangulated category algebraic if it admits a differential graded enhancement (i.e., an enrichment in complexes of abelian groups). Certainly, there is a notion of a tensor product on ...
Mikhail Bondarko's user avatar
7 votes
0 answers
241 views

A "slice-map" type problem for symmetric tensors in the square of a nuclear C*-algebra

Throughout: let $\otimes$ denote the minimal (i.e. spatial) $\newcommand{\Cst}{{\rm C}^*}\Cst$-tensor product of two $\Cst$-algebras. Let $B$ be a unital, nuclear $\Cst$-algebra and let $A\subset B$ ...
Yemon Choi's user avatar
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7 votes
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233 views

Is there a tensor norm that preserves Rosenthal Banach spaces?

By a Rosenthal Banach space I mean one that does not contain an isomorphic copy of $\ell_1$. Is there a tensor norm $\alpha$ such that the Banach tensor product $E\otimes_\alpha F$ is Rosenthal if $E$ ...
Tomás Ibarlucía's user avatar
7 votes
0 answers
316 views

tensor products NOT iterated

3-fold tensor products are usually presented in terms of the natural isomorphism of iterated tensor porducts. Where is there a treatment of 3-fold tensor products without reference to 2-fold?
Jim Stasheff's user avatar
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6 votes
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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 $X$,...
Lukas Miaskiwskyi's user avatar
6 votes
0 answers
145 views

A variant of the capset problem

Let $p > 2$ be a prime of bounded size. Suppose $A$ is a subset of $G = \mathbf{F}_p^n$ with only degenerate solutions to $$x + y = 2a,\\ x+z = 2b,\\ y + z = 2c,$$ where a solution is considered ...
Sean Eberhard's user avatar
6 votes
0 answers
283 views

When is every element of a coend of abelian groups contained in one of the summands?

Let $I$ be a small category and let $D : I^{\mathrm{op}} \times I \to \mathsf{Ab}$ be a functor. The coend $$\int^{i \in I} D(i,i)$$ can be constructed as the direct sum $\bigoplus_{i \in I} D(i,i)$ ...
Martin Brandenburg's user avatar
6 votes
0 answers
2k views

The symmetric power of a tensor product

In the representation theory, if $S^{\lambda}(V)$ is the irreductible representation of $\text{GL}(V)$ associated to a partition $\lambda \vdash n$ (in perticular, $S^n(V)$ is the $n^{\text{th}}$ ...
eti902's user avatar
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6 votes
0 answers
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Projective and injective tensor product

It is well known that for arbitrary Banach spaces $X$ and $Y$ we have that the dual space $(X \hat{\otimes}_{\pi} Y)^* = \mathcal{L}(X, Y^*)$. If we take $\ell^p$ and $\ell^q$ such that $p < q^{\...
5 votes
0 answers
159 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
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5 votes
0 answers
109 views

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}(\...
J_P's user avatar
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5 votes
0 answers
84 views

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 = ...
Dominic van der Zypen's user avatar
5 votes
0 answers
129 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 ...
Yury Korolev's user avatar
5 votes
0 answers
75 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: ...
Florian Adler's user avatar
5 votes
0 answers
102 views

Tensor square of duals over a domain

The title is motivated by my needs ($M=N$ in the sequel). Linked to the question here and there (in the case of products) is the following. Let $M,N$ $k$-modules ($k$ a commutative ring), then we ...
Duchamp Gérard H. E.'s user avatar
5 votes
0 answers
639 views

Jacobson radical of a tensor product

Let $R$ be a commutative ring and $A_1$, $A_2$ be $R$-algebras. Is there any general mean to compute the Jacobson radical of the tensor product $A_1\otimes_R A_2$ in terms of ${\rm Rad}(A_i )$, $i=1,2$...
Paul Broussous's user avatar
5 votes
0 answers
361 views

A tensor product for dg-categories

For a finite group denote by $\mathbf{Ch}^G$ the dg-category of $G$-representations in chain complexes over a field. Is there a tensor product $\otimes$ of dg-categories (similar to the Deligne ...
Lukas Woike's user avatar
  • 1,372
5 votes
0 answers
79 views

Isospectrality, Gassmann-Sunada triples, and tensor products

It is well known that Gassmann-Sunada group triples can be used to construct isospectral manifolds, arithmetically equivalent number fields, etc. (Recall that a Gassmann-Sunada triple $(U,V,W)$ ...
THC's user avatar
  • 4,313
5 votes
0 answers
207 views

Tensors and Nuclear/Fredholm Operators

For a locally convex Hausdorff spaces $E$, consider the canonical map $$\overline{\psi}:E^\prime \hat{\otimes}_\pi E \longrightarrow L(E_\sigma)$$ that maps the projective tensor product to the space ...
Matthias Ludewig's user avatar
5 votes
0 answers
267 views

Maximality of the maximal tensor product of C*-algebras

Given two C*-algebras $A$ and $B$, the maximal tensor product $A\otimes_{max}B$ is bigger than the minimal tensor product $A\otimes_{min}B$ in the sense that there exists an epimorphism $$A\otimes_{...
Ruy's user avatar
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5 votes
0 answers
601 views

Question about norm of projective tensor product

Let $A,B$ be two Banach algebra and $A\hat{\otimes} B$ be their projective tensor product. We know that every element $m$ of $A\hat{\otimes} B$ has the following representation: $$m=\sum_{n=1}^\infty ...
Hamid Shafie Asl's user avatar
4 votes
0 answers
104 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
139 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
  • 2,263
4 votes
0 answers
120 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
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4 votes
0 answers
229 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
  • 1,065
4 votes
0 answers
158 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)\...
Andromeda's user avatar
  • 189
4 votes
0 answers
112 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 $\...
gigi's user avatar
  • 1,333
4 votes
0 answers
131 views

non-abelian tensor products of several groups

R. Brown and J-L. Loday had defined the tensor product of two arbitrary groups acting on each other. Let $G,H$ be groups with actions on each other on the right. each group act on itself by ...
M masa's user avatar
  • 479
4 votes
0 answers
236 views

Exterior tensor product of $D$ Modules

The exterior tensor product of sheaves of modules is defined as: $M \boxtimes N = p_1^{*}M \otimes_{\mathcal{O}_{X \times Y}} p_2^{*}N \cong \mathcal{O}_{X \times Y} \otimes_{p_1^{-1}\mathcal{O}_X \...
Federico Barbacovi's user avatar
4 votes
0 answers
207 views

An automorphism of a tensor product commuting with $\mathrm{Id}\otimes g$

Let $V$ and $W$ be vector spaces over $\mathbb{C}$, and suppose that an automorphism $f$ of $V\otimes W$ commutes with all automorphisms of the form $\mathrm{Id}_V\otimes g$, where $g \in \mathrm{Aut}(...
Y. S's user avatar
  • 59
4 votes
0 answers
91 views

Simultaneous representations of elements of projective tensor products

Let $E,F$ be Banach spaces and consider the projective tensor product $E \widehat\otimes F$. If $\tau \in E \widehat\otimes F$ with $\|\tau\|<1$ then by definition we can find $(x_n)\subseteq E$ ...
Matthew Daws's user avatar
  • 18.5k
4 votes
0 answers
173 views

Tensor product of bornological spaces and linear functionals

It is easy to see that the dual space of a bornological space $V$ (i.e. the space of bounded linear functionals) may be zero (just take any vector space with the power set as bornology). Hence in ...
Matthias Ludewig's user avatar
4 votes
0 answers
389 views

Epimorphisms between external tensor products

Let $k$ be a field, $R,S$ commutative $k$-algebras. If $\mathsf{Mod}_{fp}$ denotes the category of finitely presented modules, the external tensor product $\mathsf{Mod}_{fp}(R) \otimes_k \mathsf{Mod}_{...
Martin Brandenburg's user avatar
3 votes
0 answers
139 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
  • 1,479
3 votes
0 answers
104 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
160 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
0 answers
170 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
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
  • 493
3 votes
0 answers
65 views

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{...
Faniel's user avatar
  • 623
3 votes
0 answers
61 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} \...
Janik's user avatar
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