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0
votes
2answers
62 views

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
votes
2answers
184 views

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
votes
1answer
111 views

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 ...
-1
votes
1answer
231 views

Why does infinite tensor product associated with some vectors in the operator algebras?

I notice that in the definition of infinite tensor product in the operator algebras, such as C*-algebras and W*-algebras, every component in the product is associated with a vector(or s state) and the ...
0
votes
0answers
60 views

Bounding the absolute value of a polynomial involving a Diophantine equation

Let $\mathbf{z}\in\mathbb{C}^n$ with entries $z_1,z_2,\ldots,z_n$. I would like to bound the following quantity \begin{equation*} ...
6
votes
1answer
174 views

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 ...
2
votes
0answers
63 views

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
votes
4answers
123 views

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
votes
0answers
112 views

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 ...
9
votes
2answers
388 views

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 ...
2
votes
0answers
76 views

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, ...
0
votes
0answers
28 views

represent hankel matrix by low rank tensorial approximation

suppose that we have given following matrix \begin{matrix} x_1 & x_2 & ..x_p \\ x_2 & x_3 & ...x_{p+1} \\ . & .& . & \\ x_{N-p+1} & x_{n-p+2} &... x_n ...
4
votes
0answers
132 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 $E$ ...
7
votes
1answer
239 views

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 ...
12
votes
0answers
255 views

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 ...
17
votes
1answer
508 views

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
2answers
251 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
votes
1answer
150 views

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 ...
1
vote
0answers
237 views

Tensor product of d.g-algebras

I'd like to prove that the tensor product functor $- \otimes Y$, where $Y$ is a d.g-algebra over a field of characteristic 0, preserves finite products of d.g-algebras. This statement is in a paper by ...
2
votes
1answer
124 views

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 ...
4
votes
3answers
389 views

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 ...
3
votes
2answers
279 views

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: ...
2
votes
0answers
105 views

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 ...
3
votes
0answers
229 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 ...
6
votes
0answers
130 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 ...
4
votes
3answers
516 views

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 ...
3
votes
1answer
173 views

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
votes
1answer
295 views

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
votes
3answers
294 views

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 ...
2
votes
2answers
155 views

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 ...
3
votes
2answers
144 views

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 ...
2
votes
1answer
335 views

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 ...
2
votes
1answer
244 views

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 ...
0
votes
1answer
149 views

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 ...
12
votes
1answer
563 views

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 ...
9
votes
1answer
485 views

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 ...
10
votes
2answers
780 views

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 ...
6
votes
1answer
253 views

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, ...
7
votes
0answers
269 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?
4
votes
2answers
368 views

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
0answers
140 views

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 ...
9
votes
0answers
341 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 ...
1
vote
1answer
388 views

Left-Module Structure on the Tensor Product ofTwo Left Modules

Given a noncommutative ring $R$, and two (left) $R$-modules $M$ and $N$, how does one define a left action on the the vector space tensor product $M \otimes N$? Multiplying on just the first factor of ...
5
votes
2answers
275 views

Limit of simple tensors

I have two questions which are intuitively true. Let $V$ be a Hilbert space. As usual we can turn $V\otimes V$ or $V\otimes V\otimes V$ into Hilbert spaces by intorducing the natural inner product ...
2
votes
1answer
162 views

Changing Left Comodules into Right Comodules via the Antipode, and Comodule Tensor Products

Let $H$ be a Hopf algebra, with invertible antipode, and let $(M,\Delta_M)$ and $(N,\Delta_N)$ be two left $H$-comodules. Now as we all know, we have a left $H$-comodule structure on the tensor ...
1
vote
1answer
321 views

Associated graded of a filtration of a tensor product

I'm trying to understand a part of the PhD thesis of Kenji Lefèvre-Hasegawa (e.g. available here). My question is about the proof of Lemma 1.3.2.3b stating: Remarquons que nous avons un ...
5
votes
2answers
1k views

Tensor product of simple modules

Let $M$ a right simple module and $N$ be a left simple module over a ring $R$. I'm seeking a kind of Schur's lemma, with $\mathrm{Hom}_R (M,N)$ replaced by $M \otimes_R N$. So my questions are: Can ...
0
votes
1answer
377 views

A doubt about tensor product on Hilbert Spaces

An operator is a bounded (i.e., continuous) linear transformation between Hilbert spaces. Let $\mathcal{B}[\mathcal{H}]$ be the set of all operators in the Hilbert space $\mathcal{H}$. Let ...
5
votes
2answers
806 views

Completion and Tensor Product of Algebras

Let $A$ be a commutative ring with 1, $I$ an ideal in $A$, $B$ an $A$-algebra. I am trying to prove the following isomorphism of $A$-algebras: $$ \big( A^* \otimes _A B \big) ^* \cong B^* $$ "$^*$" ...
0
votes
1answer
273 views

Free Module with a Projective Sub- Module, and Tensor Products

Let us consider a unital algebra $A$, with a subalgebra $B \subseteq A$, along with an $A$-$A$-bimodule $M$ which is free as a right module, and a subspace $N$ (with respect to the action of the field ...