Questions tagged [tensor-products]
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412
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Does the Boardman-Vogt tensor product of operads commute with their W-construction
I have absolutely no idea whether this is true or not but it could well be useful for me in the future if it is. If we have topological operads $\mathcal{P}$ and $\mathcal{Q}$ and we let $W$ denote ...
13
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0
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321
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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 ...
4
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2
answers
378
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Topologies for which $\mathcal{M}(X)\otimes \mathcal{M}(Y)$ is dense in $\mathcal{M}(X\times Y)$
Are there complete TVS topologies for which $\mathcal{M}(X)\otimes \mathcal{M}(Y)$ is dense in $\mathcal{M}(X\times Y)$
This question is strongly linked to
is the space of all borel measures on $\...
5
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1
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When is the direct product of two graph cores itself a core?
A graph homomorphism $f$ is a function $f : V(X) \to V(Y)$ such that if $uv \in E(X)$, then $f(u)f(v) \in E(Y)$. If such an $f$ exists, write $X \to Y$. $X$ and $Y$ are hom-equivalent if $X \to Y$ and ...
6
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1
answer
298
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Making the identification $\tau M\approx TM\oplus (TM\odot TM)$
Given a smooth manifold $M$, there is a vector bundle over $M$, denoted $\tau M$, known as the second-order tangent bundle. The fiber $\tau_mM$ at $m\in M$ is the collection of linear operators $A_m:C^...
1
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1
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377
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Necessary and sufficient condition for $can : A^X\otimes_A A^Y\rightarrow A^{X\times Y}$ to be an embedding
The two sets are, of course, supposed infinite.
This question is related to that one
Commutation of tensor products with inverse limits in a specific case
where it received a (partial) answer ($A$ ...
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1
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Commutation of tensor products with inverse limits in a specific case
For $X,Y$ sets, let's denote $Y^X$ the set of all mappings $X\rightarrow Y$. If $Y(=R)$ is a ring, $R^X$ is a $R$-module (well, a bi-module but my question is - at first - concerning commutative rings)...
1
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1
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539
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Annihilator of tensor product when $R$ is domain
Let $R$ be a Noetherian domain and $M$ and $N$ be two faithful $R$-modules. Is it true that $\operatorname{Ann}_R(M\otimes_R N)=0$?
1
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0
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91
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A reasonable framework to study properties of operator $A \mapsto KAK$ on Banach space
Let $K$ be a continuous linear operator on $C[0,1]$ (more, precisely, it is a linear integral operator). Then $K$ defines a continous linear operator $\widehat K$ on $\mathcal L(C[0,1])$ by the rule
$$...
2
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1
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387
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The tensor product of admissible morphisms of semi-normed modules over a normed ring is an admissible morphism (V. G. Berkovich)
Disclaimer : I found here https://mathoverflow.net/editing-help in the spoilers paragraph that putting >! would hide following things, which was a way for me to alleviate my question's presentation by ...
5
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1
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364
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Can the method of small moments prove a bound on the norms of random trilinear forms?
If $F(v_1,\dots,v_k)$ is a $k$-linear form on $\mathbb R^n$, the norm I want to consider is
$$ ||F|| = \sup \frac{ F(v_1,\dots, v_k)}{\prod_{i=1}^k \left|\left|v_i\right|\right|} $$
where the vector ...
2
votes
2
answers
706
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Tensor product-definition-balanced versus bilinear maps
When defining tensor products $M\otimes_R N$ over a commutative ring $R$ one can use a universal property with respect to bilinear maps $M\times N\rightarrow P$, for any $R$-module $P$.
On the other ...
6
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2
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699
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Exactness of an additive left Kan extension
Let $\phi:R\to S$ be a flat ring homomorphism and consider the induced adjoint pair
$$\phi_!:R-Mod\rightleftarrows S-Mod:\phi^*,$$
where $\phi_!=(S\otimes_R -)$. The right adjoint $\phi^*$ is easily ...
0
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1
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141
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Complementation in tensor products
This question, however looks innocent, looks non-trivial to me. Suppose that $X$ and $Y$ are Banach spaces and let $\alpha$ be any reasonable cross-norm on $X\otimes Y$. Reasonable means that $\|x\...
1
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1
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965
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Why does an infinite tensor product depend on some vectors for operator algebras?
I have read that, in the definition of the infinite tensor product of operator algebras such as ${\rm C}^\ast$-algebras and ${\rm W}^\ast$-algebras, every factor in the product is associated with a ...
6
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1
answer
716
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Tensor product of certain Sobolev spaces on non-compact manifolds
Let $M$ be a non-compact Riemannian manifold of bounded geometry (i.e., its injectivity radius is uniformly positive and the curvature tensor and all its covariant derivatives are bounded in sup-norm)....
2
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0
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1k
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Tensor product of commutators vs. commutator in a tensor product
Let $R$ be a (noetherian) commutative ring, and let $V$ and $W$ be finitely generated free $R$-modules. Let $X \subseteq \mathrm{End}_R(V)$ and $Y \subseteq \mathrm{End}_R(W)$ be finite subsets, and ...
0
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4
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529
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about the structure of components of tensor product if more than one bipartite graph is taken
I was reading about tensor product of graphs. We know that if we take tensor product of n graphs and want this product to be a connected graph then at most one graph should be bipartite. In the book ...
2
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0
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272
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Inductive and projective tensor product
Does anyone know if there is a characterization of the spaces on which the inductive tensor product and the projective tensor product are the same ? This is the same as asking every separately ...
10
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2
answers
873
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Continuity of the product map
Let $A$ be a $C^*$-algebra.
Is it possible to characterize $A$ for which the product map defined by
$$\sum\limits_{i=1}^n a_i\otimes b_i \mapsto \sum\limits_{i=1}^n a_i b_i$$
is continuous with ...
3
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0
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186
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Smooth function over a manifold into an algebra
Let $M$ be a manifold and $A$ a $*$-algebra. Does is hold that
$$C^{\infty}(M,A) \cong C^{\infty}(M) \otimes A$$
where the RHS means that you take smooth functions which map into $A$. If this holds, ...
7
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0
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233
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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$ ...
14
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1
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Infinite tensor product of states
Tensor products of finite number of different objects are always well described in the literature. However, the situation of infinite tensor products seems to be much tougher.
Even in the simplest ...
14
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0
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632
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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 ...
20
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1
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846
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Is Grothendieck classification of tensor norms and Kuratowski's 14 sets theorem somehow related?
It is known that there are only 14 reasonable tensor norms in $Ban$. On the other hand it is well known fact for topologists that one can obtain only 14 different sets from a given set applying ...
0
votes
2
answers
314
views
Tensor powers of an algebra all isomorphic
Let $k$ be a commutative ring with total quotient ring $K$, and let $A$ be a commutative $k$-algebra such that the multiplication map $A \otimes_k A \longrightarrow A$ is an isomorphism.
EDIT: Assume ...
0
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1
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734
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Highest weights of irreducible components of tensor product of irreducible sl(3)-module [closed]
I am study the representation theory of $sl(3)$ and I have a question about the tensor representation of irreducible $sl(3)$-modules as follows:
For each weight $\mu$, let $L(\mu)$ be the irreducible ...
3
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1
answer
696
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Tensor product of topological abelian groups with the reals
Given an abelian group A, the tensor product $A \otimes R$, where R are the reals, is naturally an R-vector space.
Now suppose that A is a topological abelian group (if necessary, we can assume it to ...
3
votes
3
answers
911
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A non-trivial probability measure on $2^{\mathbb R}$
Consider the measurable space $2^{\mathbb R}$, equipped with the tensor-product $\sigma$-algebra. Famously, this space has a measurable structure which is not generated by a topology (see this answer)....
5
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2
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2k
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What is the correct definition of the (derived) tensor product over a dg-algebra?
Let $A_\bullet$ be a dg-algebra over a field $k$. Let $M_\bullet$ (resp. $N_\bullet$) be a right (resp. left) $A_\bullet$-module. There is then a notion of the derived tensor product:
$$M_\bullet\...
2
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0
answers
167
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Projective tensor powers of Banach spaces over a normed field
Let $E$ be a Banach space over a complete normed field $\mathbb K$. Is it possible to classify all proper $E$ for which the projective seminorm $p_n$ defined on the $n$-th tensor power of $E$ is a ...
4
votes
0
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389
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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}_{...
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427
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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 ...
5
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3
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1k
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Künneth formula for Ext groups
Setup: Let $X,Y$ be quasi-compact quasi-separated schemes defined over a field $k$. If necessary, you can also assume that $X,Y$ are noetherian, but I don't want to assume that $X,Y$ have the ...
6
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1
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288
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Does the topological Varopoulos algebra consist of functions that are continuous and Varopoulos norm bounded?
Let $X_1,\dots,X_n$ be compact Hausdorff spaces. Let's define the Varopoulos algebra as the projective tensor product: $$V(X_1,\dots,X_n) := C(X_1) \hat{\otimes} \dots \hat{\otimes} C(X_n),$$ i.e. the ...
7
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1
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473
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positive elements in tensor product
Let x be a positive element in the spatial tensor product of two non unital C* algebras
A and B. Is there a single element $a \otimes b \geq x$?
How can we noncommutativize the following proof, in ...
4
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3
answers
730
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On tensor products of "generic" vectors
Suppose that $x_1,\ldots,x_n$ are $n$ vectors in $\mathbb{R}^m$ (where $m<n\leq m^2$) such that any subset of $m$ of them are linearly independent (i.e., they are "generic"). Now, form the $m^2\...
2
votes
2
answers
275
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Equivalence relations in suplattices
I am wondering about generalisations of the concept of equivalence relations to suplattices.
Here is my motivation: Given a set $X$. The powerset $\mathcal{P}(X)$ is a suplattice. For suplattices ...
6
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2
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573
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Natural Isomorphism of $S(V[1])$ and $(\bigwedge V)[n]$
Let $V:=\oplus_{j\in\mathbb{Z}}V_j$ be a graded $\mathbb{F}$-vector space over
the field $\mathbb{F}$. The graded tensor product of graded vector spaces is given
by
$V \otimes W:= \oplus_{j\in \...
3
votes
1
answer
958
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An example of a tensor product consisting of only simple tensors?
Hy guys. I'm doing some independent analysis which makes use of the tensor product of modules (over commutative rings with unit 1, and ring homomorphisms map $1 \mapsto 1$). Let $\pi: A' \to A$ be a ...
1
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1
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653
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Tensor product of C*-algebras of bounded, uniformly continuous functions on metric spaces
This is a follow up question to this one.
If $X$ is a metric space, denote by $C_u(X)$ the $C^\ast$-algebra of all bounded, uniformly continuous functions on $X$ (with the sup-norm).
Do we have $...
1
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1
answer
186
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Is there a wedge which operates on multiple vector spaces?
Let's say I have two vector spaces $V,W$ , and we have the graded algebras $\Lambda(V),\Lambda(W)$, each with an operation $\wedge$. I'd like to know if there are "many" $\wedge$ operators, or if ...
16
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1
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Inductive tensor product and smooth functions
Given complete, locally convex Hausdorff vector spaces $E$ and $F$, let
$$ E \otimes_i F, \qquad E \otimes_\pi F$$ denote the (completed) inductive and projective tensor products respectively. The ...
13
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1
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1k
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Classification of symtrivial modules over a PID
Let us call a module $M$ over a commutative ring $R$ symtrivial if the symmetry $M \otimes M \to M \otimes M, a \otimes b \mapsto b \otimes a$ equals the identity (the same notion applies to arbitrary ...
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Is the tensor product of polyhedra a polyhedron?
Conventions: A polytope in a finite-dimensional $\mathbb R$-vector space $V$ is defined to be a convex hull of finitely many points in $V$. A polyhedron in a finite-dimensional $\mathbb R$-vector ...
8
votes
1
answer
779
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A doubt about the parts of the spectrum of tensor products
Let $\mathcal{H}$ be any complex Hilbert space of infinite dimensional. By an operator $T$ I mean a linear bounded transformation from $\mathcal{H}$ into $\mathcal{H}$, i.e, $T:\mathcal{H}\rightarrow\...
7
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316
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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?
5
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Generators of the kernel of multiplication of a ring
Is the following true?
Let $R$ be a commutative ring and $S$ be an $R$-algebra. Let $\mu: S\otimes_R S\to S$ be the multiplication.
Let $I$ be the kernel of $\mu$, and $J\subseteq I$ be the smallest ...
4
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2
answers
574
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A construction of tensor product (Coutinho's book: D-module)
I am reading the book of Coutinho: A primer of Algebraic $D$-modules. In past, I usually study commutative algebra, so I am a freshmen with non-commutative (Weyl) algebra? In Chapter 12 of the ...
1
vote
0
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464
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How does the Schmidt decomposition generalize to tensor products of several finite-dimensional systems?
Let $n,d_1,\ldots,d_n > 1$ be integers, and $V_1, \ldots, V_n$ be inner product spaces over $\mathbb C$, having dimensions $d_1, \ldots, d_n$ respectively. We consider the ways in which we may ...