On the Homology Theory of Operator Algebras

Abstract

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.

1. Introduction

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 [1] and Tsygan [2] 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 [5].

Brylinski and Nistor [6] 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 [7] studied Hochschild and cyclic homology of finite type algebras using abelian stratifications of their primitive ideal spectrum.

Victor Nistor [8] has studied associative 𝑝-summable quasi homomorphism’s and 𝑝-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.

Results of Connes [1] have led much research interest into the computation of cyclic (co)homology groups in recent years (see, [4, 6, 9–13]).

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 [3], 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 [14].

Suppose that 𝐴 and 𝐵 be commutative unital Banach algebra with involution (in short 𝐵^∗-algebra). And let 𝐶_𝑛(𝐴) denote the (𝑛+1) fold projective tensor power of 𝐴. The elements of this Banach space 𝑛-dimensional will be called chains. Let 𝑡_𝑛∶𝐶_𝑛(𝐴)→𝐶_𝑛(𝐴), 𝑛=1,2,… denote the operator uniquely defined by 𝑡_𝑛𝑎₀⊗⋯⊗𝑎_𝑛=(−1)^𝑛𝑎_𝑛⊗𝑎₀⊗⋯⊗𝑎_𝑛−1. (1.1) And let 𝐶𝐶_𝑛(𝐴) denote the quotient space of 𝐶_𝑛(𝐴) modulo the closure of the linear span of elements of the form 𝑥−𝑡_𝑛𝑥(𝑛=1,2,…). Note that Im(𝑖𝑑_{𝑐𝑛(𝐴)}−𝑡_𝑛) is closed in 𝐶_𝑛(𝐴) and so 𝐶𝐶_𝑛(𝐴)=𝐶_𝑛(𝐴)/Im(𝑖𝑑_{𝐶𝑛(𝐴)}−𝑡_𝑛) and also 𝐶𝐶₀(𝐴)=𝐶₀(𝐴)=𝐴.

Define the complex 𝐶(𝐴)=(𝐶∗(𝐴),𝛿∗), where 𝐶_𝑛(𝐴)=𝐴⊗⋯⊗𝐴 and, 𝑏^∗∶𝐶_𝑛(𝐴)→𝐶_𝑛−1(𝐴) is the boundary operator; 𝑏_𝑛𝑎₀⊗⋯⊗𝑎_𝑛=^𝑛−1_𝑖=0(−1)^𝑖𝑎₀⊗⋯⊗𝑎_𝑖𝑎_𝑖−1⊗⋯⊗𝑎_𝑛−1. (1.2)

We can easily verify that 𝑏_𝑛+1𝑏_𝑛=0 and hence Ker𝑏_𝑛⊃Im𝑏_𝑛−1. The group 𝐻_𝑛(𝐴)=𝐻(𝐶(𝐴))=(Ker𝑏_𝑛)/(Im𝑏_𝑛−1) is called the simplicial (Hochschild) homology of Banach algebra 𝐴.

Note that: Ker𝑏_𝑛 is always closed, but, in general Im𝑏_𝑛−1 is not closed.

Considering a unital Banach algebra 𝐴, one acts on the complex 𝐶(𝐴), by the cyclic group of order (𝑛+1) by means of the operator 𝑡_𝑛∶𝐶_𝑛(𝐴)→𝐶_𝑛(𝐴) which we denote.

The quotient complex 𝐶𝐶_𝑛(𝐴)=𝐶_𝑛(𝐴)/Im(1−𝑡_𝑛) is a subcomplex of the complex 𝐶_𝑛(𝐴). Following [15], the cyclic homology of a Banach algebra 𝐴 is the homology of the complex 𝐶𝐶_𝑛(𝐴).

Given commutative unital Banach algebras 𝐴 and 𝐵, let 𝑓∶𝐴→𝐵 be algebras homomorphism. We define a free resolution of a Banach algebra 𝐵 over the homomorphism 𝑓∶𝐴→−^𝑖𝑅−→^𝜋𝐵, where 𝑖 is an inclusion and 𝜋 is a quasi-isomorphism, and use this fact to define the relative cyclic homology 𝐻𝐶_∗𝐴−→^𝑓𝐵=𝐻_∗𝑅𝐴+[𝑅,𝑅]+1−𝑡_𝑛, (1.3) where [𝑅,𝑅] is the commutant of Banach algebra 𝑅.

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 𝑉=∑^∞_𝑛=0𝑉_𝑛 be a graded vector space over ring 𝐹(𝐹=𝑅or𝐶). Suppose that 𝑅 is a differential graded 𝐹-algebra and let 𝑅⟨𝑉⟩=𝑅^∗𝑇_𝑘(𝑉) be the free product of Banach algebras, where 𝑇_𝑘(𝑉)=∑_𝑗≥0𝑉^⊗𝑗 is the tensor Banach algebra over 𝐹. The product in 𝑅⟨𝑉⟩ is given by 𝑟₁𝑒₁,…,𝑟_𝑛𝑒_𝑛𝑟_𝑛+1⋅𝑟₁̂𝑒₁̂,…,𝑟_𝑘̂𝑒_𝑘̂𝑟̂_𝑘+1=𝑟₁𝑒₁,…,𝑟_𝑛𝑒_𝑛𝑟_𝑛+1𝑟₁̂𝑒₁̂,…,𝑟_𝑘̂𝑒_𝑘̂𝑟̂_𝑘+1,𝑟_𝑖,𝑟_𝑗̂∈𝑅,𝑒_𝑖𝑒_𝑗̂∈𝑇_𝑘(𝑉). (1.4)

Definition 1.2. Let 𝑓∶𝑅₁→𝑅₂ be a homomorphism of differential graded 𝐹-algebras. An algebra 𝑅₂ is a free Banach algebra over the homomorphism𝑓 if there exists an isomorphism 𝛼∶𝑅₁⟨𝑉⟩≈𝑅₂, where 𝐸 is a differential graded vector space with the following commutative diagram: (1.5) where 𝑖is an inclusion map.

Lemma 1.3. Let 𝑓∶𝐴→𝐵 be a homomorphism of 𝐹-algebras. Then there exists a differential graded Banach algebra 𝑅=∑_𝑖=0𝑅_𝑖 with the following properties.(i)𝜋 is surjection, and the following diagram is commutative: (1.6)where 𝑖 is the inclusion map.
Clearly, there is an isomorphism 𝑗∶𝑅→𝐴 such that 𝑗∘𝑖=1_𝐴.(ii)𝜋 is quasi-isomorphism, that is, 𝜋_∗∶𝐻_∗(𝐴)→𝐻_∗(𝐵)=𝐵, where 𝐵 is a differential graded Banach algebra, (𝐵)_𝑖=𝐵,𝑖=00,𝑖>0,𝑎𝑛𝑑𝑡ℎ𝑒𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑙𝜕^𝐵=0. (1.7)(iii) The differential graded Banach algebra 𝑅 is free over the homomorphism 𝑖∶𝐴→𝑅.

Proof. We proof this theorem by two steps.(1) We construct a commutative diagram of a Banach algebras (1.8)where 𝑅⁽⁰⁾ is free over the homomorphism 𝑖₀∶𝐴→𝑅⁽⁰⁾, 𝜋₀ an surjection. Define 𝐴⟨(𝑡_𝑖)⟩=𝐸(𝑡_𝑖), where 𝐸(𝑡_𝑖) is an involutive vector space generated by {𝑡_𝑖}, or generated by the family {𝑡_𝑖𝑡^∗_𝑖}. The automorphism ∗∶𝐸(𝑡_𝑖)→𝐸(𝑡_𝑖) is given as follows ^∗(𝑡_𝑖)=(𝑡^∗_𝑖)^∗(𝑡^∗_𝑖)=𝑡_𝑖.
We choose a system {ℜ_𝑖⁽⁰⁾} of generators in a Banach algebra𝐵.This family is assumed to be closed under an involutive on 𝐵.
Now, let 𝑅⁽⁰⁾=𝐴⟨𝑡_𝑖⁽⁰⁾⟩, where 𝑡_𝑖⁽⁰⁾ is equivalent to the generator {ℜ⁽⁰⁾} in a Banach algebra 𝐵, and suppose that 𝛽_𝑖⁽⁰⁾=𝑡_𝑖⁽⁰⁾ or (𝑡_𝑖⁽⁰⁾)^∗. We define 𝜋₀ using the universal property of 𝑅⁽⁰⁾. Let 𝜋₀ be the unique involutive Banach algebras 𝑅⁽⁰⁾→𝐵 homomorphism, which restricts 𝑓 on 𝐴 and sends 𝑡_𝑖⁽⁰⁾ to ℜ_𝑖⁽⁰⁾.
Since 𝑖₀∶𝐴→𝐴⟨𝑡_𝑖⁽⁰⁾⟩ is an inclusion map, 𝑖₀(𝑎)=𝑎, 𝑖₀ is a Banach algebras homomorphism, and 𝜋₀𝑖₀(𝑎)=𝜋₀(𝑎)=𝑓(𝑎). Hence, diagram (1.8) is commutative and 𝜋₀ is surjective.
Let 𝑗₀∶𝑅⁽⁰⁾→𝐴 be the unique algebras homomorphism restricting to the identity on 𝐴 and mapping 𝑡_𝑖⁽⁰⁾ to zero. 𝑅⁽⁰⁾ is a differential graded 𝐾-Banach algebra (see[13]); 𝑅⁽⁰⁾_𝑖=𝑅⁽⁰⁾,𝑖=00,𝑖>0,andthedifferential𝜕^𝑅(0)𝐵_𝑖⁽⁰⁾=0. (1.9) The algebra 𝑅⁽⁰⁾ is free over the homomorphism 𝑖₀∶𝐴→𝑅⁽⁰⁾ since 𝑅⁽⁰⁾=𝐴⟨𝑡_𝑖⁽⁰⁾⟩.(2) We construct the second commutative diagram (1.10)where 𝑅⁽¹⁾ is free over the homomorphism 𝑖₁∶𝐴→𝑅⁽¹⁾ and 𝜋₁ surjection. Choose a system {ℜ_𝑗⁽¹⁾} of generators of Ker𝜋₀, which is closed under involution. Let 𝑡_𝑖⁽¹⁾ be indeterminate which are bijection with the ℜ_𝑗⁽¹⁾. Define 𝑅⁽¹⁾=𝐴⟨𝑡_𝑖⁽⁰⁾,𝑡_𝑗⁽¹⁾⟩, where 𝑡_𝑖⁽⁰⁾ is as defined above. Suppose that 𝛽_𝑗⁽¹⁾ denotes 𝑡_𝑗⁽¹⁾ or (𝑡_𝑗⁽¹⁾)^∗. The homomorphism 𝜋₁ is defined as to be the unique algebras 𝑅⁽¹⁾→𝐵 restricting to 𝜋₀ on 𝑅⁽⁰⁾ and sending 𝑡_𝑗⁽¹⁾ to zero. As can be seen, the homomorphism 𝜋₁ can be defined as 𝜋₀ and that 𝜋₁ is surjective since 𝜋₁(𝛽_𝑖⁽⁰⁾)=ℜ_𝑖, 𝜋₁(𝛽_𝑗⁽¹⁾)=0_𝑖. The homomorphism 𝑖₁∶𝐴→𝐴⟨𝑡_𝑖⁽⁰⁾,𝑡_𝑗⁽¹⁾⟩ is inclusion. The diagram (1.10) is commutative since (𝜋₁𝑖₁)(𝑎)=𝜋₁(𝑎)=𝑓(𝑎).
The homomorphism 𝑗₁ is defined to be the unique homomorphism: 𝑅⁽¹⁾→𝐴 of involutive Banach algebras restricting to identity on 𝐴 and mapping 𝑡_𝑖⁽¹⁾ to zero. The algebra 𝑅⁽¹⁾=𝐴⟨𝑡_𝑖⁽⁰⁾,𝑡_𝑗⁽¹⁾⟩ is free over 𝑖.
Finally, we have a differential graded Banach algebra 𝑅⁽¹⁾=𝑅⁽¹⁾₀⊕𝑅⁽¹⁾₁⊕,…,deg𝛽_𝑖⁽¹⁾=0,deg𝛽_𝑗⁽¹⁾=1. (1.11) The differential 𝜕^𝑅(1)_𝑖 is the unique derivation on 𝑅⁽¹⁾ satisfying the graded Leibntiz rule and commuting with the involution which restricts to zero on 𝑅⁽¹⁾ and sends 𝑡_𝑖⁽¹⁾ to ℜ_𝑗⁽¹⁾. So 𝜕^𝑅(1)₀𝛽_𝑖⁽⁰⁾=0, 𝜕^𝑅(1)₁𝛽_𝑗⁽⁰⁾=ℜ_𝑗⁽¹⁾∈Ker𝜋₀,𝑖>1.
Similarly, we can consider the commutative diagram (1.12) where 𝑅⁽²⁾=𝐴⟨𝑡_𝑖⁽⁰⁾,𝑡_𝑗⁽¹⁾,𝑡_𝑘⁽²⁾⟩ is a differential graded Banach algebra 𝑅⁽²⁾=𝑅⁽²⁾₀⊕𝑅⁽²⁾₁⊕𝑅⁽²⁾₂⊕,…,deg𝛽_𝑖⁽⁰⁾=0,deg𝛽_𝑗⁽⁰⁾=1,deg𝛽_𝑗⁽⁰⁾=2. (1.13) The differential Banach algebra 𝑅⁽²⁾ is also defined by using a universal property and, hence, 𝜕^𝑅(2)₀𝛽_𝑖⁽⁰⁾=0,𝜕^𝑅(2)₁𝛽_𝑗⁽⁰⁾=ℜ_𝑗⁽¹⁾,𝜕^𝑅(2)₁𝛽_𝑗⁽²⁾=ℑ_𝑘⁽²⁾=0,𝑖>2. (1.14) Consequently, we can construct an involutive Banach algebra 𝑅^(𝑖), 𝑖≥0 with the following commutative diagram: (1.15) where 𝜋_𝑖 is surjection, 𝑖≥0, 𝑖_𝑛=𝑃_𝑛−1∘⋯∘𝑃₀∘𝑖_𝑜 is an inclusion map from 𝐴 to 𝑅^(𝑛), 𝑃_𝑖 is also an inclusion map from 𝑃_𝑖∶𝐴𝑡_𝑚⁽⁰⁾_𝑜,𝑡_𝑚⁽¹⁾₁,…,𝑡_𝑚^(𝑖)_𝑖to𝐴𝑡_𝑚⁽⁰⁾_𝑜,𝑡_𝑚⁽¹⁾₁,…,𝑡_𝑚^(𝑖)_𝑖,𝑡_𝑚^(𝑖+1)_𝑖+1. (1.16) Define 𝑖_𝑛=𝑞_𝑛−1∘⋯∘𝑞₀∘𝑗_𝑜, where 𝑞_𝑛 is the projection of 𝑃_𝑖.
The diagram (1.15) is commutative since 𝑖_𝑛+1(𝛽_𝑖^(𝑛))=𝜋_𝑛(𝛽_𝑖^(𝑛))=0,𝑛≥𝑜.
Define 𝑅=lim𝑅_𝑛, 𝜋=lim𝜋_𝑛, 𝑖=lim𝑖_𝑛, 𝑗=lim𝑗_𝑛. Then the differential Banach graded algebra 𝑅 satisfies the items of Lemma 1.3 since:(1) 𝜋=lim𝜋_𝑛 is surjection, the diagram (1.17) is commutative since 𝑖(𝑎)=𝑎, 𝜋(𝑎)=𝑓(𝑎)(2) 𝜋 is quasi-isomorphism of differential graded algebras (1.18) where 𝜕^𝑅_𝑖=lim𝜕^𝑅_𝑖, (𝑅)₀=ker(𝜋)₀=𝐵, Im𝜕^𝑅=ker𝜕^𝑅₀, that is, 𝐻₀(𝑅)=𝐻_𝑖(𝑅)=0. (1.19)(3) the differential graded Banach algebras 𝑅 is free over the homomorphism 𝑖∶𝐴→𝑅, since 𝑅=𝐸, 𝐸 is a vector space generated by the system: 𝑡_𝑖⁽⁰⁾₀,𝑡_𝑖⁽¹⁾₁,…,𝑡_𝑖^(𝑛)_𝑛. (1.20)

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 𝐵 over 𝑓.

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 𝑓 be a homomorphism of Banach algebras 𝐴 and 𝐵 over a field 𝐾 (𝐾 is a real or a complex number set). Let 𝑅^𝐵_𝑓 be a free resolution of Banach algebra 𝐵 over 𝑓 and, for 𝑟₁,𝑟₂∈𝑅^𝐵_𝑓, let [𝑟₁,𝑟₂]=𝑟₁,𝑟₂−(−1)^{|𝑟1||𝑟2|}𝑟₂𝑟₁ where |𝑟𝑖|=deg𝑟_𝑖, 𝑖=1,2.

Let 𝐶=[𝑅^𝐵_𝑓,𝑅^𝐵_𝑓] be the linear space generated by [𝑟₁,𝑟₂], 𝑟₁,𝑟₂∈𝑅^𝐵_𝑓.

We construct the complex 𝐶=[𝑅^𝐵_𝑓,𝑅^𝐵_𝑓]. Clearly, from the definition of 𝑅^𝐵_𝑓, that Im(1−𝑡_𝑛) is a subcomplex of 𝑅^𝐵_𝑓. We have 𝜕𝑟₁𝑟₂=𝑟₁𝑟₂−(−1)^{|𝑙𝑟1||𝑟2|}𝑟₂𝑟₁=𝜕𝑟₁𝑟₂+(−1)^|𝑟2|𝑟₁𝜕𝑟₂−(−1)^{|𝑙𝑟1||𝑟2|}𝜕𝑟₂𝑟₁−(−1)^2||𝑟𝑟₂𝜕𝑟₁=𝜕𝑟₁𝑟₂−(−1)^{|𝑟2|(|𝑟1|+1)}𝑟₂𝜕𝑟₁+(−1)^|𝑟1|𝑟₁𝜕𝑟₂−(−1)^{|𝑟1|(|𝑟2|+1)}𝜕𝑟₂𝑟₁=𝜕𝑟₁𝑟₂+(−1)^|𝑟1|𝑟₁,𝜕𝑟₂,𝜕𝑟_𝑖=𝑟₁−1,𝑖=1,2. (2.1) Then [𝑅^𝐵_𝑓,𝑅^𝐵_𝑓] is subcomplex in 𝑅^𝐵_𝑓. Therefore, the chain complex of 𝐾-module [𝑅^𝐵_𝑓,𝑅^𝐵_𝑓] is a subcomplex of 𝑅^𝐵_𝑓.

Definition 2.1. Let 𝑓∶𝐴→𝐵 be 𝐹-algebras (char 𝐹=0) homomorphism, and 𝑅^𝐵_𝑓 be a free resolution of a Banach algebra 𝐵 over 𝑓. Then the relative cyclic homology is defined as follows: 𝐻𝐶_∗𝐴−→^𝑓𝐵=𝐻_∗𝑅^𝐵_𝑓𝐴+[𝑅,𝑅]+Im1−𝑡_𝑛. (2.2)

Definition 2.2. The 𝐹-algebra 𝐴⟨𝑡⟩ generated by the elements 𝑎₀𝑡𝑎₁𝑡,…,𝑡𝑎_𝑛, 𝑛≥0 can be considered as differential graded Banach algebras by requiring that the morphism 𝐴→𝐴⟨𝑡⟩ is a morphism of differential graded algebras (𝐴 is viewed as a differential graded Banach algebra concentrated in degree 0) and the deg𝑡=1, 𝜕𝑡=0 and 𝑡^∗=𝑡.

Lemma 2.3. A Banach algebra 𝐴⟨𝑡⟩ is splitable. One has a free algebra resolution of Banach algebra 𝐵=0 over the homomorphism 𝐴→0.

Proof. Using [6] from the chain complex 𝐴⟵^𝜕𝐴𝑡𝐴⟵^𝜕𝐴𝑡𝐴𝑡⟵^𝜕⋯⟵^𝜕𝐴𝑡⋯𝑡𝐴⟵^𝜕⋯, (2.3) where 𝐴𝑡,…,𝑡𝐴 (𝑛-times) is a 𝐾-module and the boundary operator 𝜕 is given by 𝜕𝑎₀𝑡𝑎₁𝑡,…,𝑡𝑎_𝑛−1𝑡𝑎_𝑛=^𝑛−1_𝑖=0(−1)^𝑖𝑎₀𝑡𝑎₁𝑡,…,𝑡𝑎_𝑖(𝜕𝑡)𝑎_𝑖+1𝑡,…,𝑡𝑎_𝑛=^𝑛−1_𝑖=0(−1)^𝑖𝑎₀𝑡𝑎₁𝑡,…,𝑡𝑎_𝑖𝑎_𝑖+1𝑡,…,𝑡𝑎_𝑛. (2.4) Note that the differential 𝜕 in 𝐴⟨𝑡⟩ is equivalent to the operator 𝛿^′_𝑛∶𝐶_𝑛(𝐴)→𝐶_𝑛−1(𝐴), defined by 𝛿^′_𝑛𝑎₀⊗⋯⊗𝑎_𝑛=^𝑛−1_𝑖=0(−1)^𝑖𝑎₀⊗⋯⊗𝑎_𝑖𝑎_𝑖+1⊗⋯⊗𝑎_𝑛. (2.5) Following [11], the complex (𝐶_𝑛(𝐴),𝛿^′_𝑛) is splitable and so is the complex 𝐴⟨𝑡⟩, that is, 𝐻_∗(𝐴⟨𝑡⟩)=0. Therefore, Banach algebra 𝐴⟨𝑡⟩ is free resolution of the Banach algebra 𝐵=0 over the homomorphism 𝐴→0.

Lemma 2.4. The complex 𝐴⟨𝑡⟩/[𝐴,𝐴⟨𝑡⟩] is a standard simplicial (Hochschild) complex.

Proof. Consider the factor complex 𝐴⟨𝑡⟩/[𝐴,𝐴⟨𝑡⟩]. It is generated by the elements 𝑎₀𝑡𝑎₁𝑡,…,𝑡𝑎_𝑛−1𝑡, since 𝑎₀𝑡𝑎₁𝑡,…,𝑡𝑎_𝑛−1𝑡𝑎_𝑛=𝑎_𝑛𝑎₀×𝑡𝑎₁𝑡,…,𝑡𝑎_𝑛−1𝑡(mod[𝐴,𝐴⟨𝑡⟩]). (2.6) The action of the differential 𝜕 on the complex 𝐴⟨𝑡⟩/[𝐴,𝐴⟨𝑡⟩] is given by 𝜕𝑎₀𝑡𝑎₁𝑡,…,𝑡𝑎_𝑛−1𝑡𝑎_𝑛=^𝑛−1_𝑖=0(−1)^𝑖𝑎₀𝑡𝑎₁𝑡,…,𝑡𝑎_𝑖𝑎𝑎_𝑖+1𝑡,…,𝑡𝑎_𝑛+(−1)^𝑛𝑎_𝑛𝑎₀𝑡𝑎₁𝑡,…,𝑎_𝑛−1𝑡. (2.7)
Consider the complex 𝐴⟵^𝑖𝑑𝐴⟵^𝛿𝐴^𝛿⨂2⟵⋯⟵^𝛿𝐴^𝛿⨂𝑛⟵⋯, (2.8) where 𝛿 is the differential in the standard Hochschild complex (see [7, 16]). Since the space (𝐴⟨𝑡⟩/[𝐴,𝐴⟨𝑡⟩])_𝑛+1 identifies with the space 𝐴^⨂𝑛+1∶𝑎₀𝑡𝑎₁,…,𝑡𝑎_𝑛𝑡→𝑎₀⊗⋅𝑎₁⊗⋯⊗𝑎_𝑛 (2.9) and the differential in 𝐴⟨𝑡⟩/[𝐴,𝐴⟨𝑡⟩] identifies with the differential in the standard Hochschild complex, then the complex 𝐴⟨𝑡⟩/[𝐴,𝐴⟨𝑡⟩] is the Hochschild complex.

Theorem 2.5. Let 𝐴 be Banach algebra with unity and involution. Then 𝐻𝐶_𝑖(𝐴→𝐵)=𝐻𝐶_𝑖−1(𝐴), where 𝐻𝐶_𝑖(𝐴) is the cyclic homology of 𝐹-algebras (char F = 0).

Proof. Consider the factor complex: 𝐴⟨𝑡⟩/[𝐴⟨𝑡⟩,𝐴⟨𝑡⟩]+Im(1−𝑟_𝑛), such that 𝑎₀𝑡𝑎₁𝑡,…,𝑡𝑎_𝑛−1𝑡=(−1)^𝑛𝑎_𝑛𝑡𝑎₀𝑡…𝑎_𝑛−1^𝑡, (2.10) where deg𝑎₀𝑡𝑎₁𝑡,…,𝑡𝑎_𝑛−1𝑡=𝑛, deg𝑎₀𝑡𝑎₁𝑡,…,𝑡𝑎_𝑛𝑡=𝑛+1.
The cyclic homology of 𝐴⟨𝑡⟩ is the homology of the complex 𝐴⟨𝑡⟩/[𝐴⟨𝑡⟩,𝐴⟨𝑡⟩]+Im(1−𝑡_𝑛). By factoring 𝐴⟨𝑡⟩, first by the subcomplex 𝐴←0←0←⋯ and second by the subcomplex [𝐴⟨𝑡⟩,𝐴⟨𝑡⟩]+Im(1−𝑡_𝑛)], we get a homomorphism 𝐶𝐶_∗(𝐴→0)→𝐶𝐶_∗−1(𝐴), which induces an isomorphism of the cyclic one homology groups 𝐻𝐶_∗(𝐴→0)→𝐻𝐶_∗−1(𝐴).

Theorem 2.6. Let 𝑓∶𝐴→𝐵 be a homomorphism of a Banach algebras over a field 𝐾(char𝐾=0). Then the relative cyclic homology 𝐻𝐶_𝑖(𝐴−→^𝑓𝐵) does not depend on the choice of the resolution.

Proof. The homomorphism 𝑓 induces homomorphism of chain complexes 𝑓_∗∶𝐶𝐶_∗(𝐴)→𝐶𝐶_∗(𝐵), (2.11) where 𝐶𝐶_∗(𝐴) is a cyclic complex. Consider the diagram (2.12) where 𝑅^𝐵_𝑓 is defined above, and 𝑖 is an inclusion map. The idea of proof is to show that the cone of the map 𝑖 is quasi-isomorphic to an arbitrary category (see [8, 17]), to the complex 𝑅^𝐵_𝑓/[𝑅^𝐵_𝑓,𝑅^𝐵_𝑓]+Im(1−𝑡_𝑛), Since 𝐻_𝑖𝑅^𝐵_𝑓=𝐵,𝑖=00,𝑖>0. (2.13) Then the isomorphism 𝜋_∗∶𝐶𝐶_∗(𝑅^𝐵_𝑓)→𝐶𝐶_∗(𝐵) induces an isomorphism of the homology of these complexes. Since 𝑖_∗∶𝐶𝐶_∗(𝐴)→𝐶𝐶_∗(𝑅^𝐵_𝑓) is an inclusion, then 𝐻𝐶_𝑖𝐴−→^𝑓𝐵→𝐻𝐶_𝑖𝐴−−−→^𝑔∘𝑓𝐶→𝐻𝐶_𝑖𝐴→−^𝑔𝐶→𝐻𝐶_𝑖−1𝐴−→^𝑓𝐵→⋯ (2.14)𝑀(𝑖_∗)≈𝐶𝐶_∗(𝑅^𝐵_𝑓)/𝐶𝐶_∗(𝐴), where 𝑀(𝑖_∗) is a cone of 𝑖 (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 𝑀(𝑓_∗)≈𝐶𝐶_∗(𝑅^𝐵_𝑓)/𝐶𝐶_∗(𝐴).
Following [11], we have 𝐶𝐶_∗(𝑅^𝐵_𝑓)/𝐶𝐶_∗(𝐴)≈𝑅^𝐵_𝑓/𝐴+[𝑅^𝐵_𝑓,𝑅^𝐵_𝑓], where 𝐶𝐶_∗ is the Connes cyclic complex, and by using the spectral sequence 𝐸²_𝑖𝑗=^𝜀𝐻_∗(𝑍/2,𝐻_∗(𝑅^𝐵_𝑓))=^𝜀𝐻𝐶_𝑖+𝑗(𝑅^𝐵_𝑓), we have 𝐶𝐶_∗𝑅^𝐵_𝑓𝐶𝐶_∗(𝐴)≈𝑅^𝐵_𝑓/𝐴+𝑅^𝐵_𝑓,𝑅^𝐵_𝑓+Im1−𝑡_𝑛. (2.16) So, 𝑀(𝑓_∗)≈𝑅^𝐵_𝑓/𝐴+[𝑅^𝐵_𝑓,𝑅^𝐵_𝑓]+Im(1−𝑡_𝑛). Then 𝐻𝐶_𝑖(𝐴−→^𝑓𝐵) does not depend on the choice of 𝑅^𝐵_𝑓.

Theorem 2.7. Let 𝐴, 𝐵, and 𝐷 be involutive Banach algebra. Then the following sequence 𝐴−→^𝑓𝐵→−^𝑔𝐷 induces the long exact sequence of relative cyclic homology; 𝐻𝐶_𝑖𝐴−→^𝑓𝐵→𝐻𝐶_𝑖𝐴−−−→^𝑔∘𝑓𝐷→𝐻𝐶_𝑖𝐵→−^𝑔𝐷→𝐻𝐶_𝑖−1𝐴−→^𝑓𝐵→⋯. (2.17)

Proof. In Theorem 2.6, it has been proved that any homomorphism 𝑓∶𝐴→𝐵 of involutive algebra in an arbitrary category is equivalent to an inclusion 𝑖∶𝐴→𝑅^𝐵_𝑓. Then, for a sequence 𝐴−→^𝑓𝐵→−^𝑔𝐶 of involutive Banach algebra, we have the following complex: (2.18)
Consider the following sequence of mapping cones 𝑜→𝑀𝑖_∗→𝑀𝑖^′_∗→𝑀𝑖_∗∘𝑖^′_∗→𝑜. (2.19) In general, the sequence (2.17) is not exact. The composition of two morphisms will be zero. However, the cone over the morphism 𝑀(𝑖_∗)→𝑀(𝑖^′_∗) is canonically homotopy equivalent to 𝑀(𝑖_∗∘𝑖^′_∗). So, we get the following exact sequence of the relative cyclic homology 𝐻𝐶_𝑖𝐴−→^𝑓𝐵→𝐻𝐶_𝑖𝐴−−−→^𝑔∘𝑓𝐶→𝐻𝐶_𝑖𝐵→−^𝑔𝐶→𝐻𝐶_𝑖−1𝐴−→^𝑓𝐵→⋯. (2.20) In the following we give an example of the cyclic homology of tensor algebra by using the free resolution fact. Let a Banach 𝐴 be 𝐹-algebra, (char𝐹=0) and 𝑀 be 𝐴-bimodule. For a chain complex 𝑉_• of modules, consider the complex 𝑆^𝑛(𝐴,𝑉_•)=𝐴⊗_𝐴⊗𝐴^op𝑉_•^⊗(𝑘+1). If we act on 𝑆^𝑛(𝐴,𝑉_•) by the cyclic group 𝑧_𝑛+1 of order (𝑛+1) by means of automorphisms, we get 𝑡_𝑛𝑣₀⊗⋯⊗𝑣_𝑛=(−1)^𝜇𝑣_𝑛⊗𝑣₀⊗⋯⊗𝑣_𝑛−1, (2.21) where 𝜇=(deg𝑝_𝑛)(∑^𝑛−1_𝑖=0deg𝑝_𝑖).
If 𝑉_• is a free resolution of 𝐴-bimodule 𝑀, then the complex 𝑆^𝑛(𝐴,𝑉_•) can be considered as complex 𝑆^𝑛(𝐴,𝑀).

Example 2.8. Let 𝑀 be 𝐴-bimodule, where 𝐴 is 𝐾-Banach algebra, 𝑇_𝐴(𝑀) be a tensor Banach algebra and Tor^𝐴_𝑖(𝑀,𝑀)=0,𝑖>0, then 𝐻𝐶_𝑖𝑇_𝐴(𝑀)=𝐻𝐶_𝑖(𝐴)⊕^∞⊕_𝑛=0𝐻_𝑖𝑍_𝑛+1;𝑆^𝑛(𝐴,𝑀). (2.22)

Proof. Suppose 𝑉_• is a free resolution of 𝐴-bimodule 𝑀. Then according to the condition Tor^𝐴_𝑖(𝑀,𝑀)=0,𝑖>0, the space 𝑇_𝐴(𝑉_•) is a free resolution of algebra 𝑇_𝐴(𝑀) over inclusion 𝑖∶𝐴→𝑇_𝐴(𝑀). Using Theorem 2.7 the long exact sequence of relative cyclic homology of the following sequence, 𝐴→−^𝑖𝑇_𝐴(𝑀)→0, we get ⋯→𝐻𝐶_𝑖𝐴→−^𝑖𝑇_𝐴(𝑀)→𝐻𝐶_𝑖𝐴→−⁰0→𝐻𝐶_𝑖𝑇_𝐴(𝑀)→0→𝐻𝐶_𝑖−1𝐴→−^𝑖𝑇_𝐴(𝑀)→−⁰⋯. (2.23) Since 𝐴 is a direct sum of 𝑇_𝐴(𝑀), we have 0→𝐻𝐶_𝑖𝐴→−^𝑖𝑇_𝐴(𝑀)→𝐻𝐶_𝑖(𝐴)→𝐻𝐶_𝑖𝑇_𝐴(𝑀)→0 (2.24) and hence 𝐻𝐶_𝑖𝑇_𝐴(𝑀)=𝐻𝐶_𝑖(𝐴)⊕𝐻𝐶_𝑖𝐴→−^𝑖𝑇_𝐴(𝑀). (2.25) To prove the theorem we show that 𝐻𝐶_𝑖𝐴→−^𝑖𝑇_𝐴(𝑀)=^∞⊕_𝑛=0𝐻_𝑖𝑍_𝑛+1;𝑆^𝑛(𝐴,𝑀). (2.26) Clearly, 𝑇_𝐴𝑉_•𝐴+𝑇_𝐴𝑉_•,𝑇_𝐴𝑉_•+Im1−𝑡_𝑛=⊕^∞_𝑛=0𝑉_•^⊗(𝑛+1)𝑇_𝐴𝑉_•,𝑇_𝐴𝑉_•+Im1−𝑡_𝑛. (2.27) Then we have the following isomorphism: ⊕^∞_𝑛=0𝑃_•^⊗(𝑛+1)𝑇_𝐴𝑉_•,𝑇_𝐴𝑉_•+Im1−𝑡_𝑛≈⊕^∞_𝑛=0𝐴⊗_𝐴⊗𝐴^op𝑃_•^⊗(𝑛+1)Im1−𝑡_𝑛. (2.28) The homology of the chain complex: ⊕^∞_𝑛=0𝐴⊗_𝐴⊗𝐴^op𝑉_•^⊗(𝑛+1)Im1−𝑡_𝑛 (2.29) is equivalent to ^∞_𝑛⊕₌₀𝐻_𝑖𝑍_𝑛+1;𝑆^𝑛(𝐴,𝑀). (2.30) From (2.25) and (2.26), the proof is completed.