Tensor Products Of Banach Algebras Research Articles

BED property for the tensor product of Banach algebras

BED property for the tensor product of Banach algebras

On Arens regularity of projective tensor product of Schatten spaces

We discuss Arens regularity of the projective tensor product of Banach algebras consisting of Schatten class operators. More precisely, exploiting Ülger’s biregularity criterion, we prove that for any Hilbert space ℋ, the Banach algebras Sp(ℋ)⊗γ Sq (ℋ) and B(S2(ℋ))⊗γ S2(ℋ) are not Arens regular for every pair 1≤p,q≤2; whereas, when ℋ is separable and S2(ℋ) is equipped with the Schur product, we show that S2(ℋ) is completely continuous and that S2(ℋ)⊗γ S2(ℋ) is Arens regular.

Module approximate biprojectivity and module approximate biflatness of Banach algebras

In this paper, in analogy with the classical case, we introduce module approximately biprojective and module approximately biflat Banach algebras. For an inverse semigroup S with the set of idempotents E, we show that the concepts of module approximate biprojectivity and module pseudo-amenability for $$l^1(S,\omega )$$ coincide (as an $$l^1(E)$$ -module). We also find necessary and sufficient conditions for the weighted semigroup algebra $$l^{1}(S,\omega )$$ to be module approximately biprojective and module approximately biflat. Finally, we study the module cohomological properties of projective tensor product of Banach algebras.

On Homotopy Invariants of Tensor Products of Banach Algebras

We generalize results of Davie and Raeburn describing homotopy types of the group of invertible elements and of the set of idempotents of the projective tensor product of complex unital Banach algebras. We illustrate our results by specific examples.

$$\sigma$$-Biflat Banach Algebras; the Older and the New Notions

We introduce a new concept of $$\sigma$$-biflatness for Banach algebras. We show difference between this new concept and the classical one. A characterization of $$\sigma$$-biflatness is also given. As we shall see, $$\sigma$$-biflatness is closely related to the notions of $$\sigma$$-amenability and $$\sigma$$-biprojectivity. We compare our definition with the existing notion of $$\sigma$$-biflat Banach algebras in the sense of Ghorbani and Bami (Bull Iran Math Soc 39(3):507–515, 2013). In addition, $$\sigma$$-biflatness of tensor product of Banach algebras is discussed.

Derivations on the tensor product of Banach algebras

In this paper, we study the ideal amenability of tensor product of Banach algebras. Among other things, we show that the ideal amenability $\mathcal A\widehat{\otimes}\mathcal B$ implies the ideal amenability of Banach algebras $\mathcal A$ and $\mathcal B$ for which character spaces of $\mathcal A$ and $\mathcal B$ are non-empty. Finally, we provide some examples in which $\mathcal A\widehat{\otimes}\mathcal B$ is ideally amenable.

Approximate amenability of tensor products of Banach algebras

Examples constructed by the first author and Charles Read make it clear that many of the hereditary properties of amenability no longer hold for approximate amenability. These and earlier results of the authors also show that the presence of a bounded approximate identity often entails positive results. Here we show that the tensor product of approximately amenable algebras need not be approximately amenable, and investigate conditions under which A and B being approximately amenable implies, or is implied by, A⊗ˆB or A#⊗ˆB# being approximately amenable. Once again, the rôle of having a bounded approximate identity comes to the fore. Our methods also enable us to prove that if A⊗ˆB is amenable, then so are A and B, a result proved by Barry Johnson in 1996 under an additional assumption.

Permanent Weak Module Amenability of Semigroup Algebras

Abstract We employ the fact that L1(G) is n-weakly amenable for each n ≥ 1 to show that for an inverse semigroup S with the set of idempotents E, ℓ1(S) is n- weakly module amenable as an ℓ1(E)-module with trivial left action. We study module amenability and weak module amenability of the module projective tensor products of Banach algebras.

Note on Arens regularity of symmetric tensor products

We investigate symmetric regularity of sums of symmetric tensor products of Banach spaces and Arens regularity of symmetric tensor products of Banach algebras. An example for the Hilbert space is obtained.

Arens-irregularity of tensor product of Banach algebras

We introduce Banach algebras arising from tensor norms. By these Banach algebras we make Arensregular Banach algebras such that tensor product becomes irregular, where is tensor norm. Weillustrate injective tensor product, does not preserve bounded approximate identity and it is notalgebra norm.

Compact Hermitian operators on projective tensor products of Banach algebras

Let U and V be, respectively, an infinite‐ and a finite‐dimensional complex Banach algebras, and let U⊗pV be their projective tensor product. We prove that (i) every compact Hermitian operator T1 on U gives rise to a compact Hermitian operator T on U⊗pV having the properties that ‖T1‖ = ‖T‖ and sp(T1) = sp(T); (ii) if U and V are separable and U has Hermitian approximation property (HAP), then U⊗pV is also separable and has HAP; (iii) every compact analytic semigroup (CAS) on U induces the existence of a CAS on U⊗pV having some nice properties. In addition, the converse of the above results are discussed and some open problems are posed.

The greatest regular subalgebra of certain Banach algebras of vector-valued functions

Given an arbitrary commutative complex Banach algebraA, it is shown that, for various classical Banach algebras ofA-valued functions, the greatest regular subalgebra consists precisely of those functions which map into the greatest regular subalgebra ofA. The main result covers the case of continuous and differentiable functions, Lipschitz functions, and Bochner integrable functions on a locally compact abelian group. The principal tools are from the theory of tensor products of Banach algebras.

Calculating the global dimension of tensor products of Banach algebras and a generalization of Phillips' theorem

Calculating the global dimension of tensor products of Banach algebras and a generalization of Phillips' theorem

Representation theory for tensor products of Banach algebras

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.

The Injective Tensor Product of Banach Algebras which are L 1 -Spaces

The Injective Tensor Product of Banach Algebras which are <i>L</i> ₁ -Spaces

Continuous derivations of valued fields

Let L⊃K be complete ultrametric fields. Every K-linear continuous derivation of L into a Banach space over L factors through the universal K-linear continuous derivation d:L→ΩbL/K. The structure of (d, ΩbL/K) is described in detail for discrete valued fields L/K. For dense valuations there are results on: extensions of continuous derivations; topologically transcendental and algebraic elements; topological p-bases. As application one finds a result on tensor products of Banach algebras over K.

Primitive Ideals in Tensor Products of Banach Algebras.

Primitive Ideals in Tensor Products of Banach Algebras.

Tensor products of Banach algebras and harmonic analysis

Tensor products of Banach algebras and harmonic analysis

Maximal Two-Sided Ideals in Tensor Products of Banach Algebras

Maximal Two-Sided Ideals in Tensor Products of Banach Algebras

Identities in tensor products of Banach algebras

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.