Representation theory for tensor products of Banach algebras

1981 ◽  
Vol 90 (3) ◽  
pp. 445-463 ◽  
Author(s):  
T. K. Carne
Keyword(s):  
Representation Theory ◽  
Tensor Product ◽  
Banach Algebra ◽  
Banach Algebras ◽  
Tensor Products ◽  
The Subject ◽  
Algebraic Tensor Product ◽  
Natural Way

The algebraic tensor product A1⊗A2 of two Banach algebras is an algebra in a natural way. There are certain norms α on this tensor product for which the multiplication is continuous so that the completion, A1αA2, is a Banach algebra. The representation theory of such tensor products is the subject of this paper. It will be shown that, under certain simple conditions, the tensor product of two semi-simple Banach algebras is semi-simple although, without these conditions, the result fails.

scholarly journals Identities in tensor products of Banach algebras

1970 ◽  
Vol 2 (2) ◽  
pp. 253-260 ◽  
Author(s):  
R. J. Loy
Keyword(s):  
Tensor Product ◽  
Banach Algebras ◽  
Tensor Products ◽  
Complex Field ◽  
Approximate Identities ◽  
Algebraic Tensor Product

Let A1, A2 be Banach algebras, A1 ⊗ A2 their algebraic tensor product over the complex field. If ‖ · ‖α is an algebra norm on A1 ⊗ A2 we write A1 ⊗αA2 for the ‖ · ‖α-completion of A1 ⊗ A2. In this note we study the existence of identities and approximate identities in A1 ⊗αA2 versus their existence in A1 and A2. Some of the results obtained are already known, but our method of proof appears new, though it is quite elementary.

scholarly journals Some results on the projective cone normed tensor product spaces over banach algebras

2018 ◽  
Vol 38 (1) ◽  
pp. 197 ◽  
Author(s):  
Dipankar Das ◽  
Nilakshi Goswami ◽  
Vishnu Narayan Mishra
Keyword(s):  
Fixed Point ◽  
Tensor Product ◽  
Banach Algebra ◽  
Fixed Point Theorems ◽  
Banach Algebras ◽  
Iteration Scheme ◽  
Tensor Product Space ◽  
The Common ◽  
The Stability ◽  
Algebraic Tensor Product

For two real Banach algebras $\mathbb{A}_1$ and $\mathbb{A}_2$, let $K_p$ be the projective cone in $\mathbb{A}_1\otimes_\gamma \mathbb{A}_2$. Using this we define a cone norm on the algebraic tensor product of two vector spaces over the Banach algebra $\mathbb{A}_1\otimes_\gamma \mathbb{A}_2$ and discuss some properties. We derive some fixed point theorems in this projective cone normed tensor product space over Banach algebra with a suitable example. For two self mappings $S$ and $T$ on a cone Banach space over Banach algebra, the stability of the iteration scheme $x_{2n+1}=Sx_{2n}$, $x_{2n+2}=Tx_{2n+1},\;n=0,1,2,...$ converging to the common fixed point of $S$ and $T$ is also discussed here.

The injective tensor product of ℒp-algebras

1978 ◽  
Vol 83 (2) ◽  
pp. 237-242 ◽  
Author(s):  
T. K. Carne ◽  
A. M. Tonge
Keyword(s):  
Tensor Product ◽  
Banach Algebra ◽  
Banach Algebras ◽  
Injective Tensor Product ◽  
Algebraic Tensor Product

When A1 and A2 are Banach algebras, their algebraic tensor product A1 ⊗ A2 has a natural multiplication. In this paper we investigate when the condition that A1 and A2 are ℒp-spaces constrains this multiplication to extend to the injective tensor product A1A2, making it a Banach algebra.

scholarly journals Geometry of the tensor product of C*-algebras

1988 ◽  
Vol 104 (1) ◽  
pp. 119-127 ◽  
Author(s):  
D. P. Blecher
Keyword(s):  
Tensor Product ◽  
Crucial Role ◽  
Tensor Products ◽  
Operator Spaces ◽  
C Algebra ◽  
Geometrical Properties ◽  
C Algebras ◽  
Algebraic Tensor Product ◽  
Natural Way

When and ℬ are C*-algebras their algebraic tensor product ⊗ ℬ is a *-algebra in a natural way. Until recently, work on tensor products of C*-algebras has concentrated on norms α which make the completion ⊗α ℬ into a C*-algebra. The crucial role played by the Haagerup norm in the theory of operator spaces and completely bounded maps has produced some interest in more general norms (see [8; 12]). In this paper we investigate geometrical properties of algebra norms on ⊗ ℬ. By an ‘algebra norm’ we mean a norm which is sub-multiplicative: α(u.v) ≤ ≤ α(u).α(v).

scholarly journals Banach algebras with one dimensional radical

1983 ◽  
Vol 27 (1) ◽  
pp. 115-119
Author(s):  
Lawrence Stedman
Keyword(s):  
Tensor Product ◽  
Banach Algebra ◽  
Banach Algebras ◽  
Quotient Algebra ◽  
One Dimensional ◽  
Natural Mapping ◽  
Algebraic Tensor Product

A Banach algebra A with radical R is said to have property (S) if the natural mapping from the algebraic tensor product A ⊗ A onto A2 is open, when A ⊗ A is given the protective norm. The purpose of this note is to provide a counterexample to Zinde's claim that when A is commutative and R is one dimensional the fulfillment of property (S) in A implies its fulfillment in the quotient algebra A/R.

Extension of normal functionals on W*-tensor products

1975 ◽  
Vol 78 (2) ◽  
pp. 301-307 ◽  
Author(s):  
Simon Wassermann
Keyword(s):  
Representation Theory ◽  
Tensor Product ◽  
Hilbert Spaces ◽  
General Result ◽  
Von Neumann Algebras ◽  
Tensor Products ◽  
Von Neumann ◽  
Hilbert Algebras ◽  
Special Cases

A deep result in the theory of W*-tensor products, the Commutation theorem, states that if M and N are W*-algebras faithfully represented as von Neumann algebras on the Hilbert spaces H and K, respectively, then the commutant in L(H ⊗ K) of the W*-tensor product of M and N coincides with the W*-tensor product of M′ and N′. Although special cases of this theorem were established successively by Misonou (2) and Sakai (3), the validity of the general result remained conjectural until the advent of the Tomita-Takesaki theory of Modular Hilbert algebras (6). As formulated, the Commutation theorem is a spatial result; that is, the W*-algebras in its statement are taken to act on specific Hilbert spaces. Not surprisingly, therefore, known proofs rely heavily on techniques of representation theory.

scholarly journals On the Structure of Monodromy Algebras and Drinfeld Doubles

1997 ◽  
Vol 09 (03) ◽  
pp. 371-395
Author(s):  
Florian Nill
Keyword(s):  
Hopf Algebra ◽  
Tensor Product ◽  
Tensor Products ◽  
Spin Chains ◽  
Coadjoint Action ◽  
Braided Group ◽  
Gauge Transformations ◽  
Loop Algebras ◽  
Algebraic Tensor Product ◽  
Quasitriangular Hopf Algebra

We give a review and some new relations on the structure of the monodromy algebra (also called loop algebra) associated with a quasitriangular Hopf algebra H. It is shown that as an algebra it coincides with the so-called braided group constructed by S. Majid on the dual of H. Gauge transformations act on monodromy algebras via the coadjoint action. Applying a result of Majid, the resulting crossed product is isomorphic to the Drinfeld double [Formula: see text]. Hence, under the so-called factorizability condition given by N. Reshetikhin and M. Semenov–Tian–Shansky, both algebras are isomorphic to the algebraic tensor product H ⊗ H. It is indicated that in this way the results of Alekseev et al. on lattice current algebras are consistent with the theory of more general Hopf spin chains given by K. Szlachányi and the author. In the Appendix the multi-loop algebras ℒm of Alekseev and Schomerus [3] are identified with braided tensor products of monodromy algebras in the sense of Majid, which leads to an explanation of the "bosonization formula" of [3] representing ℒm as H ⊗…⊗ H.

Compact semigroups in commutative Banach algebras

1969 ◽  
Vol 66 (2) ◽  
pp. 265-274 ◽  
Author(s):  
M. A. Kaashoek ◽  
T. T. West
Keyword(s):  
Banach Algebra ◽  
Topological Semigroup ◽  
Spectral Properties ◽  
Banach Algebras ◽  
Structure Theory ◽  
Compact Semigroups ◽  
Commutative Banach Algebras ◽  
Continuous Multiplication ◽  
Natural Way

A monothetic semigroup is a topological semigroup with jointly continuous multiplication which contains a dense cyclic subsemigroup. These semi-groups arise in a natural way in the study of semi-algebras. In (4) we showed that a compact monothetic semigroup in a Banach algebra can be characterized in terms of the spectral properties of a generating element. In this paper these spectral theorems are linked with the well-known structure theory of compact semigroups.

Tensor Products of Banach Algebras*

1968 ◽  
Vol 11 (5) ◽  
pp. 691-701
Author(s):  
Boaz Natzitz
Keyword(s):  
Tensor Product ◽  
Banach Algebras ◽  
Tensor Products ◽  
Group Algebras ◽  
Commutative Banach Algebras

In [3] Gelbaum defined the tensor product A ⊗CB of three commutative Banach algebras, A, B and C and established some of its properties. Various examples are given and the particular case where A, B and C are group algebras of L.C.A. groups G, H and K respectively, is discussed there. It is shown there that if K is compact L1(G) ⊗ L1(K) L1(H) is isomorphic to where is L.C.A. 1 L (K) 1 1 if and only if L1(G) ⊗ L1(K) L1(H) is semisimple.

scholarly journals A constructive proof of Gelbaum's theorem on tensor products

1973 ◽  
Vol 8 (2) ◽  
pp. 211-214
Author(s):  
David A. Robbins
Keyword(s):  
Tensor Product ◽  
Maximal Ideal ◽  
Banach Algebras ◽  
Cartesian Product ◽  
Tensor Products ◽  
Maximal Ideal Space ◽  
Constructive Proof ◽  
Ideal Space ◽  
Commutative Banach Algebras

A constructive proof is given of Gelbaum's result that the maximal ideal space of the tensor product of commutative Banach algebras is homeomorphic to the cartesian product of the maximal ideal spaces.

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