Amenable Operators And Amenable Banach Algebras

Throughout this paper, C, N denotes the complex plane and the set of natural numbers, respectively and (?) will always denote a complex separable infinite dimen-sional Hilbert space. Let B((?)) denote the algebra of all bounded linear operators on (?). We denote by K((?)) the ideal of compact operators in B((?)). If T∈23((?)), de-notes the norm-closure of span{Tk:k∈{0}∪N} by 21T, the idempotents of 21T by P(21T), the Banach algebra which is generated by the idempotents of 21T by P(21T), the commutant of 21T by 21T', the double cominutant of 21T by 21T".The concept of amenable Banach algebras was first introduced by Johnson. Sup-pose that 21 is a Banach algebra. A Banach 21-bimodule is a Banach space X that is also an algebra 2l-bimodulc for which there exists a constant K>0 such that ||a·x||≤K||a||||x|| and ||x·a||≤K||a||||x|| for all a∈21 and x∈X. A Banach 21-bimodule X is said to be commutative if a·x= x·a for each a∈21, x∈X. We note that X*, the dual of X, is a Banach 21-bimodule with respect to the dual actions Such a Banach 21-bimodule is called a dual 21-bimodule.A derivation D:21→X is a continuous linear map such that D(ab)= a·D(b)+ D(a)·b, for all a, b∈21. Given x∈X, the inner derivationδx:21→X, is defined byδx(a)=a·x-x·a.According to Johnson's original definition, a Banach algebra 21 is amenable if every derivation from 21 into the dual 21-bimodule X* is inner for all Banach 2l-bimodules X. As a complement to this notion, a Banach algebra 21 is contractible if every derivation from 21 into every Banach 21-bimodule is inner; is weak amenable if every derivation from A into every dual.A-bimodule is inner.Ever since its introduction, the concept of amenability has occupied an important place in the research of Banach algebras, operator algebras and harmonic analysis. For example, an early result of Johnson shows that the amenability of the group algebra L1(G), for G a locally compact group, is equivalent to the amenability of the underlying group G. Results of Connes and IIaagerup show that a C*-algebra is amenable if and only if it is nuclear. However, the structure of amenable Banach algebras are far less understood.One of the first result in this direction is due to Seinberg, he proved that ifΩis a compact Hausdorff space and if 21 (?) C(Ω) is an amenable uniform algebra that separates points, then 21= C(Ω). Subsequently, Curtis and Loy extension Seinberg's result to the non-commutative Banach algebras. They prove that if 21 (?) B((?)) is an amenable Banach algebra and if 21 is generated by its normal elements, then 21 is a C* algebra. In view of those results, Seinberg, Curtis, Loy and Khelemskii raised the following conjecture for the structure of amenable Banach algebras:Conjecture SCLK A Banach subalgebra of (?) ((?)) is amenable if and only if it is similar to a C*-algebra.In 1995, Willis studied the amenability of the Banach algebras which is generated by an operator on IIilbert space. An operator T∈B((?)) is called to be an amenable(or, weakly amenable) operator, if 21T is an amenable (respectively, weakly amenable) Ba-nach algebra. He showed that if T is a compact amenable operator, then T is similar to a normal operator. Subsequently, Gifford extension the result to the subalgebras of K((?)) and prove that if a norm-closed algebra 21 (?) K((?)) is amenable, then 21 is similar to a C* algebra.In 2005, Farenick, Forrest and Marcoux study the amenability (weak amenability) of normal operator and the amenability of triangular operator. They showed that if T is similar to a normal operator, then 21T is amenable if and only if 21T is weakly amenable if and only if 21T is similar to a C*-algebra and the spectrum of T has connected complement and empty interior; if T is a triangular operator with respect to an orthonormal basis of (?), then 21T is amenable if and only if T is similar to a normal operator whose spectrum has connected complement and empty interior. In view of those results, they raised the following problem:Problem FFM Assume that T∈B((?)) is a compact and weakly amenable operator, does T similar to a normal operator?According to the results of Curtis, Loy and Farenick, Forrest, Marcoux, it is not difficult to see that, Conjecture SCLK is equivalent to every amenable operator is similar to a normal operator, in the case of the Banach algebra is generated by an operator.The first part of this thesis mainly deals with the structure of amenable operators on complex separable infinite dimensional Hilbert space. At first, we use the reduction theory of von Neumann to give two equivalent descriptions for Conjecture SCLK. We obtain the following result:Theorem 0.1 The following are equivalent:(i)If an amenable Banach algebra is generated by an operator, then the algebra is similar to C* algebra;(ii) Every non-scalar amenable operator has non-trivial hyperinvariant subspace;(iii) For every amenable operator T, there exists an invertible operator X such that 21XTX-1" is a reductive algebra and T has non-trivial invariant subspace;(iv) Every amenable operator is similar to a normal operator.Then, we give the characterization of the structure of amenable operators in cases of invariant and hyperinvariant subspaces, respectively.Theorem 0.2 Assume that T∈B((?)) is an amenable operator, there exists two hy-perinvariant subspaces M1,M2 of T, such that T has the decomposition T=T1+T2, respective the space decomposition (?)= M1+M2 and satisfy:(i) T1,T2 are amenable operators and T1 similar to normal operator; (ii) M1 is the largest hyperinvariant subspace of T, such that T|M1 is similar to a normal operator;(iii) For any hyperinvariant subspace M of T2, T2|M can not similar to a normal oper-ator;(iv) P(21T2)'' similar to a von Neumann algebra which has the uniform multiplicity∞;(v)(?)2∩K(M2)={0}.Theorem 0.3 Assume that T∈B((?)) is an amenable operator, there exists two invari-ant subspaces N1,N2 of T, such that T has the decomposition T=A1+A2, respective the space decomposition (?)= N1+N2 and satisfy:(i) A1,A2 are amenable operators and A1 similar to normal operator;(ii) N1 is the largest invariant subspace of T, such that T|N1 is similar to a normal operator;(iii) For any hyperinvariant subspace N of A2, A2|N can not similar to a normal oper-ator;(iv)P(21A2)" similar to a von Neumann algebra which has the uniform multiplicity∞,If the answer to Conjecture SCLK is positive, by theorem 0.1, every amenable is similar to a normal operator. Then, for the above theorem M1=N1=(?). That is to say, the two decompositions of theorem 0.2 and 0.3 are the same. In fact, we show that the two decompositions are really the same which supporting Conjecture SCLK.Based on the results above, we obtain the following result which gives a, negative answer to FFM problem:Theorem 0.4 For setσ={0,λ1,λ2,…}, where is a sequence of positive real numbers which converge to zero, there exists a compact operator T with spectrumσ, and T is weakly amenable and character amenable but not similar to a normal operator. It is well known, the problem of invariant and hyper invariant subspaces is still an open problem. But, it has been proved that there exists non-trivial invariant or hyperinvariant subspaces for some special class operators. Hence, at last, we describe the amenability for some special class operators and obtained the following results:Theorem 0.5 If T∈B((?)) and satisfies any conditions of the following, then T is amenable if and only if T is similar to a normal operator whose spectrum has connected complement and empty interior:(i) T is a subnormal operator or an n-normal operator for some n∈N;(ii) T=B1B2, here, B1,B2 are positive operators;(iii) T is approximately similar or quasi-similar to some normal or compact operator;(iv) There exists a compact operator K in (?)T with nul K<∞;(v) Every densely defined graph transformation for 21T is bounded;(vi) T=N+Q, where N is a normal operator, Q is a polynomial compact operator and NQ= QN;(vii) T=N+K, where N is a normal operator whose essential spectrum has only finite accumulation point and K is a compact operator;(iix) T=N+K, where N is a normal operator and exists p>1 such that K∈Cp andσ-(T)∪σ(N) contained in some smooth Jordan arc,(ix) There exists p>1 such thatThe other part of our research focuses on the generalized notions of character amenability. We introduce (uniformly) approximately character amenable (contrac-tive) Banach algebra. At first, we characterize the structure of generalized notions of character amenable Banach algebras in different ways.Theorem 0.6 For a Banach algebra 21 the following are equivalent: (i) 21 is (uniformly) approximately character amenable;(ii) has (bounded, respectively) both left and right approximate identities, and for any for all a∈21 (uniformly on the unit ball of 21, respectively);(iii) 21 has (bounded, respectively) both left and right approximate identities, and for any (uniformly on the unit ball of 21, respectively).Theorem 0.7 For a Banach algebra 21 the following are equivalent:(i) 21 is (uniformly) approximately character contractible;(ii) 21 has (an identity, respectively) both right and left approximate identities and for anyφ∈σ(21), there exist nets all a∈21 (uniformly on the unit ball of 21, respectively);(iii) 21 has (an identity, respectively) both right and left approximate identities and for (uniformly on the unit ball of 21, respectively).Based on the results above, we describe the relations among generalized notions of character amenability completely as follows: where CC denotes character contractibility, UACC denotes uniform approximate char-acter contractibility, ACC denotes approximate character contractibility, CA denotes character amenability, UACA denotes uniform approximate character amenability and AC A denotes approximate character amenability. A→B or A J B denotes A=> B but B (?) A; A(?)B or A(?)B denotes A and B are equivalent.