scholarly journals Tensor products of positive definite quadratic forms, V

1981 ◽  
Vol 82 ◽  
pp. 99-111 ◽  
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Tensor Product ◽  
Quadratic Forms ◽  
Tensor Products ◽  
Positive Definite ◽  
List Type

Our aim is to proveTHEOREM. Let L be a positive lattice of E-type such that [L: L̃] < ∞ and L̃ is indecomposable. (i)If L ≅ L1 ⊗ L2for positive lattices L1, L2, then L1, L2 are of E-type and [L1: L̃1], [L2:L̃2] < ∞ and L̃1, L̃2 are indecomposable.(ii)If L is indecomposable with respect to tensor product, then for each indecomposable positive lattice X we have(1)L ⊗ X ≅ L ⊗ Y implies X ≅ Y for a positive lattice Y,(2)If X= ⊗t L ⊗ X′ where X′ is not divided by L, then O(L ⊗ X) is generated by O(L), O(X′) and interchanges of L’s, and(3)L ⊗ X is indecomposable.

scholarly journals Tensor Products of Positive definite Quadratic forms III

1978 ◽  
Vol 70 ◽  
pp. 173-181 ◽  
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Quadratic Forms ◽  
Tensor Products ◽  
Positive Definite ◽  
List Type ◽  
Quadratic Lattices

In the previous papers [2], [3] we treated the following two questions. Let L,M,N be positive definite quadratic lattices over Z: (i) If L, M are indecomposable, then is L⊗M indecomposable?(ii) Does L ⊗ M ⋍ L ⊗ N imply M ⋍ N?

scholarly journals Tensor products of positive definite quadratic forms IV

1979 ◽  
Vol 73 ◽  
pp. 149-156 ◽  
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Quadratic Forms ◽  
Tensor Products ◽  
Positive Definite ◽  
Orthogonal Groups ◽  
Quadratic Lattices

Let L, M, N be positive definite quadratic lattices over Z. We treated the following problem in [5], [6]:If L ⊗ M is isometric to L ⊗ N, then is M isometric to N?We gave a condition (**) in [6] such that the answer is affirmative for an indecomposable lattice L satisfying (**), and we gave some examples. In this paper we introduce a certain apparently weaker condition (A) than the condition (**), and we show that the condition (A) implies the condition (**) and more on integral orthogonal groups than a result in [6].

scholarly journals Tensor products of positive definite quadratic forms, VII

1984 ◽  
Vol 96 ◽  
pp. 133-137 ◽  
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Quadratic Form ◽  
Lie Algebras ◽  
Quadratic Forms ◽  
Tensor Products ◽  
Positive Definite ◽  
Positive Definite Quadratic Form ◽  
Constant Multiple ◽  
The Third ◽  
Cartan Matrices

In this paper we generalize results of the third paper of this series. As a corollary we can show the following: Let Li (1 ≤ i ≤ n) be a positive definite quadratic form which is equivalent to one of Cartan matrices of Lie algebras of type An (n ≥ 2), Dn (n ≥ 4), E6, E7, E8 and assume that is positive definite quadratic forms and satisfies that rk Mt ≥ 2 and implies rk K or rk L = 1. Then we have n = m and Lt is equivalent to a constant multiple of Ms(i) for some permutation s. Therefore we get the uniqueness of decompositions with respect to tensor products in this case.

scholarly journals On the tensor product of quaternion algebras of characteristic two

1988 ◽  
Vol 30 (1) ◽  
pp. 111-113
Author(s):  
P. Mammone
Keyword(s):  
Tensor Product ◽  
Quadratic Forms ◽  
Sufficient Conditions ◽  
Tensor Products ◽  
Necessary And Sufficient Conditions ◽  
Division Algebras ◽  
Quaternion Algebras ◽  
Characteristic Two ◽  
Necessary And Sufficient

The purpose of this note is to generalize to fields of characteristic two the results obtained in [4]. We obtain necessary and sufficient conditions involving quadratic forms for certain tensor products of quaternion algebras to be division algebras.

scholarly journals The dual of the space of bounded operators on a Banach space

2021 ◽  
Vol 8 (1) ◽  
pp. 48-59
Author(s):  
Fernanda Botelho ◽  
Richard J. Fleming
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Banach Spaces ◽  
Dual Space ◽  
Tensor Products ◽  
Bounded Operators ◽  
Projective Tensor Product ◽  
Projective Tensor

Abstract Given Banach spaces X and Y, we ask about the dual space of the 𝒧(X, Y). This paper surveys results on tensor products of Banach spaces with the main objective of describing the dual of spaces of bounded operators. In several cases and under a variety of assumptions on X and Y, the answer can best be given as the projective tensor product of X ** and Y *.

scholarly journals SUBPROJECTIVITY OF PROJECTIVE TENSOR PRODUCTS OF BANACH SPACES OF CONTINUOUS FUNCTIONS

2021 ◽  
pp. 1-14
Author(s):  
R.M. CAUSEY
Keyword(s):  
Tensor Product ◽  
Banach Spaces ◽  
Tensor Products ◽  
Continuous Functions ◽  
Hausdorff Spaces ◽  
Projective Tensor Product ◽  
Projective Tensor ◽  
Spaces Of Continuous Functions

Abstract Galego and Samuel showed that if K, L are metrizable, compact, Hausdorff spaces, then $C(K)\widehat{\otimes}_\pi C(L)$ is c0-saturated if and only if it is subprojective if and only if K and L are both scattered. We remove the hypothesis of metrizability from their result and extend it from the case of the twofold projective tensor product to the general n-fold projective tensor product to show that for any $n\in\mathbb{N}$ and compact, Hausdorff spaces K1, …, K n , $\widehat{\otimes}_{\pi, i=1}^n C(K_i)$ is c0-saturated if and only if it is subprojective if and only if each K i is scattered.

FINITENESS RESULTS FOR REGULAR DEFINITE TERNARY QUADRATIC FORMS OVER ${\mathbb Q}(\sqrt{5})$

2007 ◽  
Vol 03 (04) ◽  
pp. 541-556 ◽  
Author(s):  
WAI KIU CHAN ◽  
A. G. EARNEST ◽  
MARIA INES ICAZA ◽  
JI YOUNG KIM
Keyword(s):  
Quadratic Form ◽  
Number Field ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Integral Quadratic Form ◽  
Ring Of Integers ◽  
Ternary Quadratic Forms ◽  
Totally Positive ◽  
Finiteness Result

Let 𝔬 be the ring of integers in a number field. An integral quadratic form over 𝔬 is called regular if it represents all integers in 𝔬 that are represented by its genus. In [13,14] Watson proved that there are only finitely many inequivalent positive definite primitive integral regular ternary quadratic forms over ℤ. In this paper, we generalize Watson's result to totally positive regular ternary quadratic forms over [Formula: see text]. We also show that the same finiteness result holds for totally positive definite spinor regular ternary quadratic forms over [Formula: see text], and thus extends the corresponding finiteness results for spinor regular quadratic forms over ℤ obtained in [1,3].

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