Research Article | Open Access
On the Homology Theory of Operator Algebras
We investigate the cyclic homology and free resolution effect of a commutative unital Banach algebra. Using the free resolution operator, we define the relative cyclic homology of commutative Banach algebras. Lemmas and theorems of this investigation are studied and proved. Finally, the relation between cyclic homology and relative cyclic homology of Banach algebra is deduced.
Many years ago, cyclic homology has been introduced by Connes and Tsygan and defined on suitable categories of algebras, as the homology of a natural chain complex and the target of a natural Chern character from topological (or algebraic) K-Theory.
In order to extend the classical theory of the Chern character to the noncommutative setting, Connes  and Tsygan  have developed the cyclic homology of associative algebras. Recently, there has been increasing interest in general algebraic structures than associative algebras, characterized by the presence of several algebraic operations. Such structures appear, for example, in homotopy theory [3, 4] and topological field theory .
Brylinski and Nistor  have extended Conne’s computation of the cyclic cohomology groups of smooth algebras arising from foliations with separated graphs and explained some results of Atiyah and Segal on orbifold Euler characteristic in the setting of cyclic homology. Kazhdan  studied Hochschild and cyclic homology of finite type algebras using abelian stratifications of their primitive ideal spectrum.
Victor Nistor  has studied associative -summable quasi homomorphism’s and -summable extensions elements in a bivariant cyclic cohomology group defined by Connes, and showed that this generalizes his character on K-homology; furthermore, he studied the properties of this character and showed that it is compatible with analytic index.
A promising approach to the calculation of cyclic cohomology groups is to break it down by making use of extensions of Banach algebras; this is a standard device in the study of various properties of Banach algebras.
The Banach cyclic (co)homology of Banach algebra has been studied by Christensen and Sinclair , Helemskii [4, 9], among others. The dihedral cohomology in Banach category and its relation with the cyclic cohomology, the triviality, and nontriviality of dihedral cohomology groups of some classes of operator algebras have been studied .
Suppose that and be commutative unital Banach algebra with involution (in short -algebra). And let denote the fold projective tensor power of . The elements of this Banach space -dimensional will be called chains. Let , denote the operator uniquely defined by And let denote the quotient space of modulo the closure of the linear span of elements of the form . Note that is closed in and so and also .
Define the complex , where and, is the boundary operator;
We can easily verify that and hence . The group is called the simplicial (Hochschild) homology of Banach algebra .
Note that: is always closed, but, in general is not closed.
Considering a unital Banach algebra , one acts on the complex , by the cyclic group of order by means of the operator which we denote.
The quotient complex is a subcomplex of the complex . Following , the cyclic homology of a Banach algebra is the homology of the complex .
Given commutative unital Banach algebras and , let be algebras homomorphism. We define a free resolution of a Banach algebra over the homomorphism , where is an inclusion and is a quasi-isomorphism, and use this fact to define the relative cyclic homology where is the commutant of Banach algebra .
Definition 1.1. A graded Banach algebra is a Banach algebra that has a graded normed algebra as a dense subalgebra.
We discuss the existence of the free Banach algebra resolution. Let be a graded vector space over ring . Suppose that is a differential graded -algebra and let be the free product of Banach algebras, where is the tensor Banach algebra over . The product in is given by
Definition 1.2. Let be a homomorphism of differential graded -algebras. An algebra is a free Banach algebra over the homomorphism if there exists an isomorphism , where is a differential graded vector space with the following commutative diagram: (1.5) where is an inclusion map.
Lemma 1.3. Let be a homomorphism of -algebras. Then there exists a differential graded Banach algebra with the following properties.(i) is surjection, and the following diagram is commutative:
(1.6)where is the inclusion map.
Clearly, there is an isomorphism such that .(ii) is quasi-isomorphism, that is, , where is a differential graded Banach algebra, (iii) The differential graded Banach algebra is free over the homomorphism .
Proof. We proof this theorem by two steps.(1) We construct a commutative diagram of a Banach algebras
(1.8)where is free over the homomorphism , an surjection. Define , where is an involutive vector space generated by , or generated by the family . The automorphism is given as follows .
We choose a system of generators in a Banach algebra.This family is assumed to be closed under an involutive on .
Now, let , where is equivalent to the generator in a Banach algebra , and suppose that or . We define using the universal property of . Let be the unique involutive Banach algebras homomorphism, which restricts on and sends to .
Since is an inclusion map, , is a Banach algebras homomorphism, and . Hence, diagram (1.8) is commutative and is surjective.
Let be the unique algebras homomorphism restricting to the identity on and mapping to zero. is a differential graded -Banach algebra (see); The algebra is free over the homomorphism since .(2) We construct the second commutative diagram (1.10)where is free over the homomorphism and surjection. Choose a system of generators of , which is closed under involution. Let be indeterminate which are bijection with the . Define , where is as defined above. Suppose that denotes or . The homomorphism is defined as to be the unique algebras restricting to on and sending to zero. As can be seen, the homomorphism can be defined as and that is surjective since , . The homomorphism is inclusion. The diagram (1.10) is commutative since .
The homomorphism is defined to be the unique homomorphism: of involutive Banach algebras restricting to identity on and mapping to zero. The algebra is free over .
Finally, we have a differential graded Banach algebra The differential is the unique derivation on satisfying the graded Leibntiz rule and commuting with the involution which restricts to zero on and sends to . So , .
Similarly, we can consider the commutative diagram (1.12) where is a differential graded Banach algebra The differential Banach algebra is also defined by using a universal property and, hence, Consequently, we can construct an involutive Banach algebra , with the following commutative diagram: (1.15) where is surjection, , is an inclusion map from to , is also an inclusion map from Define , where is the projection of .
The diagram (1.15) is commutative since .
Define , , , . Then the differential Banach graded algebra satisfies the items of Lemma 1.3 since:(1) is surjection, the diagram (1.17) is commutative since , (2) is quasi-isomorphism of differential graded algebras (1.18) where , , , that is, (3) the differential graded Banach algebras is free over the homomorphism , since , is a vector space generated by the system:
Definition 1.4. The differential graded Banach algebra which satisfies the conditions (i), (ii), and (iii) of Lemma 1.3 is called a free resolution of Banach algebra over .
2. The Relative Cyclic Homology of Banach Algebra
In this part, we define the relative cyclic homology of commutative unital Banach algebra and study its properties. Let be a homomorphism of Banach algebras and over a field ( is a real or a complex number set). Let be a free resolution of Banach algebra over and, for , let where , .
Let be the linear space generated by , .
We construct the complex . Clearly, from the definition of , that is a subcomplex of . We have Then is subcomplex in . Therefore, the chain complex of -module is a subcomplex of .
Definition 2.1. Let be -algebras (char ) homomorphism, and be a free resolution of a Banach algebra over . Then the relative cyclic homology is defined as follows:
Definition 2.2. The -algebra generated by the elements , can be considered as differential graded Banach algebras by requiring that the morphism is a morphism of differential graded algebras ( is viewed as a differential graded Banach algebra concentrated in degree 0) and the , and .
Lemma 2.3. A Banach algebra is splitable. One has a free algebra resolution of Banach algebra over the homomorphism .
Proof. Using  from the chain complex where (-times) is a -module and the boundary operator is given by Note that the differential in is equivalent to the operator , defined by Following , the complex is splitable and so is the complex , that is, . Therefore, Banach algebra is free resolution of the Banach algebra over the homomorphism .
Lemma 2.4. The complex is a standard simplicial (Hochschild) complex.
Proof. Consider the factor complex . It is generated by the elements , since
The action of the differential on the complex is given by
Consider the complex where is the differential in the standard Hochschild complex (see [7, 16]). Since the space identifies with the space and the differential in identifies with the differential in the standard Hochschild complex, then the complex is the Hochschild complex.
Theorem 2.5. Let be Banach algebra with unity and involution. Then , where is the cyclic homology of -algebras (char F = 0).
Proof. Consider the factor complex: , such that
where , .
The cyclic homology of is the homology of the complex . By factoring , first by the subcomplex and second by the subcomplex , we get a homomorphism , which induces an isomorphism of the cyclic one homology groups .
Theorem 2.6. Let be a homomorphism of a Banach algebras over a field . Then the relative cyclic homology does not depend on the choice of the resolution.
Proof. The homomorphism induces homomorphism of chain complexes
where is a cyclic complex. Consider the diagram
where is defined above, and is an inclusion map. The idea of proof is to show that the cone of the map is quasi-isomorphic to an arbitrary category (see [8, 17]), to the complex , Since
Then the isomorphism induces an isomorphism of the homology of these complexes. Since is an inclusion, then
, where is a cone of (see [12, 15, 18]).
Note that, the symbol denotes a quasi-isomorphism. It is clear, from the above discussion, that the following diagram is commutative (2.15) and hence .
Following , we have , where is the Connes cyclic complex, and by using the spectral sequence , we have So, . Then does not depend on the choice of .
Theorem 2.7. Let , , and be involutive Banach algebra. Then the following sequence induces the long exact sequence of relative cyclic homology;
Proof. In Theorem 2.6, it has been proved that any homomorphism of involutive algebra in an arbitrary category is equivalent to an inclusion . Then, for a sequence of involutive Banach algebra, we have the following complex:
Consider the following sequence of mapping cones In general, the sequence (2.17) is not exact. The composition of two morphisms will be zero. However, the cone over the morphism is canonically homotopy equivalent to . So, we get the following exact sequence of the relative cyclic homology In the following we give an example of the cyclic homology of tensor algebra by using the free resolution fact. Let a Banach be -algebra, and be -bimodule. For a chain complex of modules, consider the complex . If we act on by the cyclic group of order by means of automorphisms, we get where
If is a free resolution of -bimodule , then the complex can be considered as complex .
Example 2.8. Let be -bimodule, where is -Banach algebra, be a tensor Banach algebra and , then
Proof. Suppose is a free resolution of -bimodule . Then according to the condition , the space is a free resolution of algebra over inclusion . Using Theorem 2.7 the long exact sequence of relative cyclic homology of the following sequence, , we get Since is a direct sum of , we have and hence To prove the theorem we show that Clearly, Then we have the following isomorphism: The homology of the chain complex: is equivalent to From (2.25) and (2.26), the proof is completed.
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Copyright © 2012 Alaa Hassan Noreldeen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.