29Banach limits with values in B(\mathcal{H})
Ryotaro Tanaka

1Introduction

This note is a survey on [11]. A Banach limit is a special linear functional on the boundedsequence space \ell_{\infty} introduced as a generalization of the (limit” functional. The precise
definition is as follows,

Definition 1.1. A boundcd linear functional \varphi on \ell_{\infty} is called a Banach limit if(i) \varphi is positive, that is, if \alpha_{n}\geq 0 for each n\in \mathbb{N} then \varphi((\alpha_{n})_{n\in N})\geq 0 ;(ii) \varphi((\alpha_{n+1})_{n\in \mathbb{N}})=\varphi((\alpha_{n})_{n\in N}) for each (\alpha_{n})_{n\in N}\in\ell_{\infty} : and(iii) \varphi((\alpha_{n})_{n\in \mathbb{N}})=\lim_{n}\alpha_{n} whenever (\alpha_{n})_{n\in \mathbb{N} is a convergent sequence.
In particular, the positivity of \varphi shows that \Vert\varphi\Vert=\varphi(1)=1 . The existence of Banach
limits is guaranteed by simple arguments using the Hahn‐Banach theorem or ultrafilters
on \mathb {N} ; and such functionals have various applications in functional analysis.
On the other hand, there are some ideas to generalize the notion of Banach limits tothe case of vector sequences. Such ’operators” were initially introduced by Deeds [4] for
Hilbert spaces. The definition adopted here is found in [2, 5, 6] and bit different from thescalar case, but it is in fact equivalent to the original one for scalar sequences (that is,the case of X=\mathbb{C} in the below).
Definition 1.2. LetX

be a Banach space. Then a bounded linear operator

Tfiom

P_{\infty}(X) (the space of bounded sequences in X equipped with the supremum norm) into X
is called a Banach limit on \ell_{\infty}(X) if(i) \Vert T\Vert=1 ;(ii) T((x_{n+1})_{n\in \mathbb{N}})=T((x_{n})_{n\in \mathbb{N}}) for each (x_{n})_{n\in \mathbb{N}}\in\ell_{\infty}(X) ; and(iii) T((x_{n})_{n\in \mathbb{N}})= \lim_{n}x_{n} whenever (x_{n})_{n\in \mathbb{N} is a convergent sequence in X.
Unlike the scalar case, the first problem is their existence. Actually, there are Banach

spaces with no Banach limits. A typical example is (real) c_{0} ([2]). However, the dual
Banach spaces always have Banach limits. Indeed, for a scalar‐valued Banach limit \varphi,

putting (x, \overline{\varphi}((f_{n})_{n\in \mathbb{N} )\rangle=\varphi((f_{n}(x)) _{n\in \mathbb{N} ) for each (f_{n})_{n\in \mathbb{N}}\in\ell_{\infty}(X^{*}) and each x\in X yieldsan X^{*} ‐valued Banach limit \overline{\varphi}.

For the case of \mathcal{B}(\mathcal{H}) , the space of bounded linear operators on a Hilbert space \mathcal{H} , one
has another natural way to introduce Banach limits. For a scalar‐valued Banach limit\varphi , let \langle\overline{\varphi}_{\mathcal{H} ( A_{n})_{n\in \mathbb{N} )x, y } =\varphi((\langle A_{n}x, y\rangle)_{n\in \mathbb{N}}) . Then \overline{\varphi}_{\mathcal{H} is a Banach limit on \ell_{\infty}(\mathcal{B}(\mathcal{H}) .

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We here emphasize that, though \overline{\varphi}_{\mathcal{H} is formally related to the Hilbert space \mathcal{H} , it is notessential. Indeed, if we consider \mathcal{B}(\mathcal{H}) as the dual space (of the trace class), then \overline{\varphi}_{\mathcal{H} =\overline{\varphi}
since \rho(\overline{\varphi}_{\mathcal{H}}((A_{n})_{n\in N}))=\varphi((\rho(A_{n}))_ {n\in N}) holds for each ultraweakly continuous functionals\rho on \mathcal{B}(\mathcal{H}) . Hence the property of \overline{\varphi}_{\mathcal{H} seems to depend on the operator algebraic structureof \mathcal{B}(\mathcal{H}) rather than the operator theoretic one.
In this paper, we show that every \mathcal{B}(\mathcal{H}) ‐valued Banach limit has the form \overline{\varphi}_{\mathcal{H} for some
Banach limit \varphi on \ell_{\infty} . As an application, it is shown that the notion of almost convergencefor sequences in \mathcal{B}(\mathcal{H}) cannot be characterized by using vector‐valued Banach limits unless\mathcal{H} is finite dimensional.

2Banach limits with values in C^{*} ‐algebras

We first present the following proposition containing some basic properties of Banachlimits with values in C^{*} ‐algebras.
Proposition 2.1. Let \mathfrak{A} be a C^{*} ‐algebra. Suppose that T is a Banach limit on P_{\infty}(\mathfrak{A}) .
Then the following hold:(i)

T is positive.

(ii) T((ba_{n})_{n\in N})=bT((a_{n})_{n\in \mathbb{N}}) and T((a_{n}b)_{n\in \mathbb{N}})=T((a_{n})_{n\in \mathbb{N}})b hold for each (a_{n})_{n\in \mathbb{N} \in
P_{\infty}(\mathfrak{A}) and each b\in \mathfrak{A}.
As an immediate consequence of the preceding proposition, we have a condition on
C^{*} ‐algebras necessary for the existence of Banach limit with values in those algebras.

Corollary 2.2. Let \mathfrak{A} be a C^{*} ‐algebra. If there exists a Banach limit T on \ell_{\infty}(\mathfrak{A}) , then\mathfrak{A} is monotone a‐complete, that is, every norm bounded monotone increasing sequence in\mathfrak{A} has a least upper bound.

For a detailed investigation on monotone a‐completeC^{*} ‐algebras,

the readers arereferred to the monograph of Saitô and Wright [10].
Problem 2.3. What conditions on C^{*} ‐algebras \mathfrak{A} are necessary and sufficient for theexistence of \mathfrak{A} ‐valued Banach limits?

3Banach limits on P_{\infty}(\mathcal{B}(\mathcal{H})We begin this section with the main result in this paper which shows that the set ofvector‐valued Banach limits with values in \mathcal{B}(\mathcal{H}) is in a one‐to‐one correspondence withthat of usual complex‐valued Banach limits.

Theorem 3.1. Let \mathcal{H} be a Hilbert space. If T is a Banach limit on \ell_{\infty}(\mathcal{B}(\mathcal{H}) , then
T=\overline{\varphi}_{\mathcal{H} for some Banach limit \varphi on \ell_{\infty}.
We here note that \mathcal{B}(\mathcal{H}) ‐valued Banach limit \overline{\varphi}_{\mathcal{H} is restricted to any von Neumann
algebra \mathcal{R} acting on \mathcal{H} . Indeed, for each (A_{n})_{n\in \mathbb{N} \in\ell_{\infty}(\mathcal{R}) and each A'\in \mathcal{R}' , we have\langle\overline{\varphi}_{\mathcal{H}}((A_{n})_{n\in N})A'x, y\rangle=\varphi( (\{A_{n}A'x, y\rangle)_{n\in N})=\varphi((\{A_{n}x, (A')^{*}y\rangle)_{n\in \mathbb{N} )=\langle A'\overline{\varphi}_{\mathcal{H} ((A_{n})_{n\in \mathbb{N} )x, y\},31
31which shows that \overline{\varphi}_{\mathcal{H} ( A_{n})_{n\in \mathbb{N} )A'= A'\overline{\varphi}_{\mathcal{H} ( A_{n})_{n\in N}) , that is, \overline{\varphi}_{\mathcal{H} ( A_{n})_{n\in \mathbb{N} )\in \mathcal{R}"= \mathcal{R} by
the double commutant theorem.

The following results provide some special properties of Banach limits \overline{\varphi}_{\mathcal{H} with values
in von Neumann algebras \mathcal{R} acting on \mathcal{H}.

Proposition 3.2. Let \mathcal{R} be a von Neumann algebra acting on a Hilbert space \mathcal{H} , and let\varphi be a Banach limit on \ell_{\infty} . Then \rho(\overline{\varphi}_{\mathcal{H} ((A_{n})_{n\in N}))=\varphi((\rho(A_{n}))_ {n\in \mathbb{N} ) for each (A_{n})_{n\in \mathbb{N}}\in\ell_{\infty}(\mathcal{R}) and each \rho\in \mathcal{R}_{*}.

Corollary 3.3. Let \mathcal{R} and \mathcal{S} be von Neumann algebras acting on Hilbert spaces \mathcal{H} and\mathcal{K} ,respectively. Suppose that \Phi : \mathcal{R}arrow S is a*‐isomorphism, and that \varphi is a Banachlimit on \ell_{\infty} . Then

\Phi^{-1}(\overline{\varphi}_{\mathcal{K} ( \Phi(A_{n}) _{n\in \mathbb{N} ) = \overline{\varphi}_{\mathcal{H} ( A_{n})_{n\in \mathbb{N} )for each (A_{n})_{n\in \mathbb{N} \in\ell_{\infty}(\mathcal{R}) .

In what follows, for a von Neumann algebra \mathcal{R} and a Banach limit \varphi on \ell_{\infty} , let \overline{\varphi}
denote the \mathcal{R} ‐valued Banach limit satisfying \rho(\overline{\varphi}((A_{n})_{n\in \mathbb{N}}))=\varphi((\rho(A_{n}))_{n\in N}) for each
(A_{n})_{n\in \mathbb{N} \in\ell_{\infty}(\mathcal{R}) and each \rho\in \mathcal{R}_{*} . By Proposition 3.2, if \mathcal{R} acts on a Hilbert space \mathcal{H},then \overline{\varphi}=\overline{\varphi}_{\mathcal{H} .

We wonder whether Banach limits with values in von Neumann algebras has a general

form.

Problem 3.4. Can we classify, or determine a general form of Banach limits with values
in von Neumann algebras?

The following problem might be the first step.
Problem 3.5. Let \mathcal{R} be a von Neumann algebra, and let T be a Banach limit with values
in \mathcal{R} . Suppose that T satisfies T((\alpha_{n}I)_{n\in N})\in \mathbb{C}I for each (\alpha_{n})_{n\in \mathbb{N} \in\ell_{\infty} . Does T havethe form \overline{\varphi} for some Banach limit \varphi on \ell_{\infty} ?

4Almost convergence in \mathcal{B}(\mathcal{H})The notion of almost convergence can be found in Lorentz [9] and Boos [3]; see also [1,2, 5, 6]. A sequence (x_{n})_{n\in \mathbb{N} in a Banach space X is said to be almost convergent to anelement x\in X (called the almost limit of (x_{n})_{n\in \mathbb{N} ) if

1\dot{ \imath} m\sup_{m\in\mathb {N} p\Vertp^{-1}\sum_{j=0}^{p-1}x_{m+j}- x\Vert=0.
In the scalar case, almost convergence is characterized by using scalar‐valued Banachlimits. Namely, if (\alpha_{n})_{n\in \mathbb{N} \in\ell_{\infty} and \alpha\in \mathbb{C} , then \alpha is the almost limit of (\alpha_{n})_{n\in \mathbb{N} if andonly if \varphi((\alpha_{n})_{n\in \mathbb{N} )=\alpha for each Banach limit \varphi on \ell_{\infty} . However, in the case of vector
sequences, only one‐sided implication is known, that is, if x is the almost ıimit of (x_{n})_{n\in \mathbb{N}then T((x_{n})_{n\in \mathbb{N}})=x for each X ‐valued Banach limit. This can be shown by a simple

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argument essentially found in the proof of [2, Theorem 2]. Indeed, if S is the unilateralshift on \ell_{\infty}(X) given by S((x_{n})_{n\in N}) ) =(x_{n+1})_{n\in \mathbb{N}} , then it follows from

\lim_{p}\sup_{m\in \mathb {N} \Vert p^{-1}\sum_{j=0}^{p-1}x_{m+j}-x\Vert=0
that p^{-1} \sum_{J^{=0}}^{p-1}S^{j}( x_{n})_{n\in \mathbb{N} ) ) converges to(x, x, . . .) in \ell_{\infty}(X) as parrow\infty . Hence we haveT( x_{n})_{n\in \mathb {N} )=1 \dot{ \imath} mTp(p^{-1}\sum_{j=0}^{p-1}S^{j} ( x_{n})_{n\in \mathb {N} ) =x
for each Banach limit T on \ell_{\infty}(X) . On the other hand, whether the converse holds true
is depend on case by case; see [4, 7, 8], for related results. A Banach space X is said toverify the vector‐valued version of the Lorentz theorem if x is the almost limit of (x_{n})_{n\in \mathbb{N}
whenever T((x_{n})_{n\in \mathbb{N}})=x for each X ‐valued Banach limit T.
As a consequence of Theorem 3.1, we have the following result which provides a natural
Banach space that does not verify the vector‐valued version of the LorentL theorem.

Theorem 4.1. Let \mathcal{H} be a Hilbert space. Then \mathcal{B}(\mathcal{H}) verifies the vector‐valued version ofthe Lorentz theorem if and only if \mathcal{H} is finite dimensional.

Remark 4.2. The convergence with respect to \mathcal{B}(\mathcal{H}) ‐valued Banach limits is just the*weak version of almost convergence. Indeed, a sequence (A_{n})_{n\in \mathbb{N} in \mathcal{B}(\mathcal{H}) and an element
A\in \mathcal{B}(\mathcal{H}) satisfies T((A_{n})_{n\in \mathbb{N}})=A for each \mathcal{B}(\mathcal{H}) ‐valued Banach limit if and only if\overline{\varphi}((A_{n})_{n\in \mathbb{N}})=A for each Banach limit \varphi in \ell_{\infty} by Theorem 3.1 and Lemma 3.2, whichhappens if and only if \rho(A)=\varphi((\rho(A_{n}))_{n\in \mathbb{N}}) for each \rho\in \mathcal{B}(\mathcal{H})_{*} and each Banach limit\varphi in \ell_{\infty} . Finally, this last statement just means \rho(A) is the almost limit of (\rho(A_{n}))_{n\in \mathbb{N}for each \rho\in \mathcal{B}(\mathcal{H})_{*}.

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[8] J. C. Kurtz, Almost convergent vector sequences, Tôhoku Math. J., 22 (1970), 493‐498.

[9] G. G. Lorentz, A contribution to the theory of divergent sequences, Acta Math., 80(1948), ı67‐190.

[10] K. Saitô and J. D. M. Wright, Monotone complete C^{*} ‐algebras and generic dynamics,
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[11] R. Tanaka, On vector‐valued Banach limits with values in \mathcal{B}(\mathcal{H}) , to appear in Publ.
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Ryotaro Tanaka
Faculty of Mathematics,
Kyushu University,
Fukuoka 819‐0395, JapanE‐mail: r‐tanaka@math.kyushu‐u.ac.jp(Current address)
Faculty of Industrial Science and Technology,
Tokyo University of Science,
Hokkaido 049‐3514, JapanE‐mail: r‐tanaka@rs.tus.ac jp