Banach-Alaoglu theorem for Hilbert $H^*$-module

TL;DR

This paper extends the Banach-Alaoglu theorem to Hilbert H*-modules by constructing a suitable weak* topology, demonstrating the compactness of the unit ball in this new setting.

Contribution

It introduces a novel Banach-Alaoglu type theorem for Hilbert H*-modules, establishing a weak* topology under which the unit ball is compact.

Findings

Established a $\Lambda$-weak$^*$ topology on Hilbert H*-modules

Proved the unit ball is compact in this topology

Extended classical Banach-Alaoglu theorem to a new algebraic setting

Abstract

We provided an analogue Banach-Alaoglu theorem for Hilbert $H^*$-module. We construct a $\Lambda$-weak$^*$ topology on a Hilbert $H^*$-module over a proper $H^*$-algebra $\Lambda$, such that the unit ball is compact with respect to $\Lambda$-weak$^*$ topology.

Equations

$$L_{w_0^{\prime },x_1,..,x_n,S_1,..,S_n,\delta }=\left\{w^{\prime }\in L^{\prime }|\left|\mathrm{sp}(S_j(w^{\prime }(x_j)-w_0^{\prime }(x_j)))\right|<\delta ,j=1,..,n\right\},$$

$$S_a(\lambda x)=S(a\lambda x)=\lambda S(ax)=\lambda S_a(x),S_a(xy)=S(axy)=S(ax)y=S_a(x)y$$

$$||S_a(x)||=||S(ax)||\le ||S||\cdot ||ax||\le ||S||\cdot ||a||\cdot ||x||,$$

$$||(L_{a_{n}a}-S_a)(x)||$$

$$=||L_{a_n}(ax)-S(ax)||\le ||L_{a_n}-S||\cdot ||ax||$$

$$\le ||L_{a_n}-S||\cdot ||a||\cdot ||x||,$$

$$K=\left\{w^{\prime }\in L^{\prime }|||w^{\prime }(x)||\le 1\text{for all}x\in B_1(L)\right\}$$

$$P=\{f:L\to \tau (\Lambda )|||f(x)||\le ||x||\text{for all}x\in L\}.$$

$$W_1=\{w^{\prime }\in L^{\prime }|$$

$$x_1,x_2,...,x_n\in L,S_1,...,S_n\in C(\Lambda )\}$$

$$W_2=\{f\in P|$$

$$x_1,x_2,...,x_n\in L,S_1,...,S_n\in C(\Lambda )\}$$

$$\left|\mathrm{sp}(S[a(w^{\prime }(x)-w_0^{\prime }(x))])\right|<\epsilon ,\left|\mathrm{sp}(S[b(w^{\prime }(y)-w_0^{\prime }(y))])\right|<\epsilon ,$$

$$\left|\mathrm{sp}(S[(w^{\prime }(xa+yb)-w_0^{\prime }(xa+yb))])\right|<\epsilon .$$

$$|\mathrm{sp}(S(w_0^{\prime }(xa+yb)-w_0^{\prime }(x)a-w_0^{\prime }(y)b))|$$

$$=|\mathrm{sp}(S[(w_0^{\prime }(xa+yb)-w^{\prime }(xa+yb))-(w_0^{\prime }(x)a-w^{\prime }(x)a)$$

$$-(w_0^{\prime }(y)b-w^{\prime }(y)b)])|$$

$$=|\mathrm{sp}(S(w_0^{\prime }(xa+yb)-w^{\prime }(xa+yb))-S[(w_0^{\prime }(x)-w^{\prime }(x))a]$$

$$-S[(w_0^{\prime }(y)-w^{\prime }(y))b])|$$

$$\le \left|\mathrm{sp}(S(w_0^{\prime }(xa+yb)-w^{\prime }(xa+yb)))\right|+\left|\mathrm{sp}(S[(w_0^{\prime }(x)-w^{\prime }(x))a])\right|$$

$$+\left|\mathrm{sp}(S[(w_0^{\prime }(y)-w^{\prime }(y))b])\right|$$

$$=\left|\mathrm{sp}(S(w_0^{\prime }(xa+yb)-w^{\prime }(xa+yb)))\right|+\left|\mathrm{sp}(aS[(w_0^{\prime }(x)-w^{\prime }(x))])\right|$$

$$+\left|\mathrm{sp}(bS[(w_0^{\prime }(y)-w^{\prime }(y))])\right|$$

$$\le \epsilon +\epsilon +\epsilon =3\epsilon .$$

$$\mathrm{sp}(S(w_0^{\prime }(xa+yb)))=\mathrm{sp}(S(w_0^{\prime }(x)a)+S(w_0^{\prime }(y)b))$$

$$\mathrm{sp}(S(w_0^{\prime }(xa+yb)-w_0^{\prime }(x)a-w_0^{\prime }(y)b))=0$$

$$\mathrm{sp}((w_0^{\prime }(xa+yb)-w_0^{\prime }(x)a-w_0^{\prime }(y)b){)}^{\ast }(w_0^{\prime }(xa+yb)-w_0^{\prime }(x)a-w_0^{\prime }(y)b))=0.$$

$$|\mathrm{sp}(S(w_0^{\prime }(x)))|<|\mathrm{sp}(S(w^{\prime }(x)))|+\epsilon \le ||S||\cdot ||w^{\prime }(x)||+\epsilon \le ||S||+\epsilon .$$

$$||w_0^{\prime }(x)||$$

$$\le \underset{S\in C(\Lambda ),||S||=1}{\sup }||S||+\epsilon \le 1+\epsilon$$

$$W_{x_0,y_1,...,y_n,S_1,...,S_n,\delta }=\left\{x\in L|\left|\mathrm{sp}(S_j([x,y_j]-[x_0,y_j]))\right|<\delta ,j=1,...,n\right\},$$

Full text

Keywords: $H^{*}$-algebra, $H^{*}$-module, compact set.

MSC(2010): Primary: 46H25; Secondary: 57N17, 46A50.

∗Corresponding author

††copyright: ©0: Iranian Mathematical Society

1. Introduction

A Hilbert $H^{*}$-module $L$ over an $H^{*}$-algebra $\Lambda$ is a right $\Lambda$-module which possesses a $\tau(\Lambda)$-valued product, where $\tau(\Lambda)=\{ab|\,\,a,b\in\Lambda\}$ is the trace-class. At the same time, $H$ is a Hilbert space with the inner product given by the action of the trace on the $\tau(\Lambda)$-valued product.

The notion of $H^{*}$-module is introduced by Saworotnow in [8] under the name of generalized Hilbert space. It has been studied by Smith [10], Giellis [4] Molnar [6], Cabrera [3], Martinez et al. [3], Bakić and Guljaš [2] and others. Saworotnow has proved that the trace-class in a proper $H^{*}$-algebra $\Lambda$ has pre-dual. For Hilbert $H^{*}$-modules, a generalization of Reisz theorem holds, i.e. for each bounded $\Lambda$-linear functional on $L$, there is $x_{f}\in L$ such that $f(x)=[x_{f},x]$ for all $x\in L$.

Paschke [7] showed that self-dual Hilbert $W^{*}$-modules are dual Banach spaces and found topology on module such that the unit ball is compact.

In the present paper we find a topology on Hilbert $H^{*}$-module $L$ over a proper $H^{*}$-algebra $\Lambda$ such that the unit ball in $L$ is compact with respect to this topology.

2. Basic notations and preliminary results

We recall that an $H^{*}$-algebra is a complex associative Banach algebra with an inner product $\left<\cdot,\cdot\right>$ such that $\left<a,a\right>=||a||^{2}$ for all $a\in\Lambda$, and for each $a\in\Lambda$ there exists some $a^{*}\in\Lambda$ such that $\left<ab,c\right>=\left<b,a^{*}c\right>$ and $\left<ba,c\right>=\left<b,ca^{*}\right>$ for all $b,c\in\Lambda$. The adjoint $a^{*}$ of $a$ need not be unique (see [1]). Proper $H^{*}$-algebra $\Lambda$ is an $H^{*}$-algebra which satisfies $a\Lambda=0\Rightarrow a=0$ (or $\Lambda a=0\Rightarrow a=0$). An $H^{*}$-algebra $\Lambda$ is proper if and only if each $a\in\Lambda$ has a unique adjoint $a^{*}\in\Lambda$ (see [1, Theorem 2.1]). An $H^{*}$-algebra is simple algebra if it has no nontrivial closed two-sided ideals.

The trace-class in a proper $H^{*}$-algebra $\Lambda$ is defined as the set $\tau(\Lambda)=\{ab|\,\,a,b\in\Lambda\}$. The trace-class is selfadjoint ideal of $\Lambda$ and it is dense in $\Lambda$, with norm $\tau(\cdot)$. The norm $\tau$ is related to the given norm $||\cdot||$ on $\Lambda$ by $\tau(a^{*}a)=||a||^{2}$ for all $a\in\Lambda$. There exists a continuous linear form $\mathrm{sp}$ on $\tau(\Lambda)$ (trace) satisfying $\mathrm{sp}(ab)=\mathrm{sp}(ba)=\left<a^{*},b\right>.$ In particular, $\mathrm{sp}(a^{*}a)=\mathrm{sp}(aa^{*})=\left<a,a\right>=||a||^{2}=\tau(a^{*}a)$.

A Hilbert $\Lambda$-module is a right module $L$ over a proper $H^{*}$-algebra $\Lambda$ provided with a mapping $[\cdot,\cdot]\colon L\times L\rightarrow\tau(\Lambda)$, which satisfies the following conditions: $[x,\alpha y]=\alpha[x,y];$ $[x,y+z]=[x,y]+[x,z];$ $[x,ya]=[x,y]\,a;$ $[x,y]^{*}=[y,x]$; $L$ is a Hilbert space with the inner product $\left<x,y\right>=\mathrm{sp}\left([x,y]\right)$ for all $\alpha\in\mathbb{C},$ $x,y,z\in L,$ $a\in\Lambda$ and for all $x\in L,$ $x\neq 0$ there is $a\in\Lambda,$ $a\neq 0$ such that $[x,x]=a^{*}a.$

A $\Lambda$-linear functional on $L$ is a mapping $w^{\prime}\colon L\rightarrow\tau(\Lambda)$ such that $w^{\prime}(xa+yb)=w^{\prime}(x)a+w^{\prime}(y)b$ for all $x,y\in L,$ $a,b\in\Lambda.$ It is bounded if there exists $c>0$ such that $\tau(w^{\prime}(x))\leq c||x||$ for all $x\in L$. In this case we define $||w^{\prime}||=\mathop{\rm supp}\nolimits\{c|\,\,\tau(w^{\prime}(x))\leq c||x||\,\,\text{for all}\,\,x\in L\}$. The norm space of all bounded $\Lambda$-linear functional $f$ on $L$ we denoted by $L^{\prime}$.

Let $R(\Lambda)$ be the set of all bounded linear operators $S$ on $\Lambda$ such that $S(xy)=(Sx)y$ for all $x,y\in\Lambda$ and let $C(\Lambda)$ be the closed subspace of $R(\Lambda)$ generated by the operators of the form $La\colon x\rightarrow ax,$ $a\in\Lambda.$

We now state some theorems which will be necessary for the proof of main results.

Theorem 2.1**.**

[8, Theorem 3]** Each bounded $\Lambda$-linear functional $f$ on $L$ is of the form $[x_{f},\cdot]$ for some $x_{f}\in L$.

Theorem 2.2**.**

[9, Lemma 1]** If $a\in\Lambda$ then the mapping $f_{a}\colon S\rightarrow\mathrm{sp}(Sa)$, defined on $C(\Lambda)$, is a bounded linear functional and $||f_{a}||=\tau(a)$.

Theorem 2.3**.**

[9, Theorem 1]** Each bounded linear functional on $C(\Lambda)$ is of the form $f_{a}$ for some $a\in\tau(\Lambda)$. The correspondence $a\leftrightarrow f_{a}$ is an isometric isomorphism between $\tau(\Lambda)$ and $C(\Lambda)^{*}$. Also, $\tau(\Lambda)$ is a Banach algebra.

For more details, we refer to [8, 9, 1, 2, 5].

3. Results

Let $L$ be a Hilbert $H^{*}$-module over a proper $H^{*}$-algebra $\Lambda$ and let $B_{1}(L)$ is the unit ball in $L.$ We construct a topology on $W^{\prime}$ such that the unit ball in $L^{\prime}$ is compact with respect to this topology. Define $\Lambda^{\prime}$-weak∗ topology on $L^{\prime}$ with the base

[TABLE]

for $w^{\prime}_{0}\in L^{\prime},$ $x_{j}\in L,$ $S_{j}\in C(\Lambda),$ and $\delta>0.$

The main result of this paper, the compactness of the unit ball, will be proven in Theorem 3.2 and its corollary. Before that, we state and prove a useful lemma.

Lemma 3.1**.**

Let $\Lambda$ be a $H^{*}$-algebra and let $S\in C(\Lambda)$. Then the operator $S_{a}\colon\Lambda\rightarrow\Lambda,$ $S_{a}(x)=S(ax)$ belongs to $C(\Lambda)$.

Proof.

From

[TABLE]

and

[TABLE]

it follows that $S_{a}$ belongs to $R(\Lambda)$. If $L_{a_{n}}$ converges to $S$, then from

[TABLE]

it follows that $L_{a_{n}a}$ converges to $S_{a}$. Thus $S_{a}\in C(\Lambda)$. ∎

Theorem 3.2**.**

The set

[TABLE]

is compact in $\Lambda^{\prime}$-weak∗ topology.

Proof.

The neighborhood $B_{1}(L)$ is absorbing because for each $x\in L,$ $x\neq 0$, there exists a number $\gamma(x)=||x||>0$ such that $x\in\gamma(x)B_{1}(L)$. For all $w^{\prime}\in K$, it holds $w^{\prime}(x)\leq 1,$ $x\in B_{1}(L)$, hence $|w^{\prime}(x)|\leq||x||$, $x\in L$.

According to Banach-Alaoglu theorem and Theorem 2.3, the set $D_{x}=\{a\in\Lambda|||a||\leq\gamma(x)\}$ is compact in weak∗-topology on $\tau(\Lambda)$ given by seminorms $p_{S}(\cdot)=|\mathrm{sp}(S(\cdot))|,$ $S\in C(\Lambda)$. Let $\tau_{1}$ be the product weak∗-topology on $P=\prod_{x}D_{x}$, the cartesian product of all $D_{x}$, one for each $x\in L$. Since each $D_{x}$ is weak∗-compact, it follows, from Tychonoff’s theorem, that $P$ is $\tau_{1}$-compact.

From definition of $P$ we have

[TABLE]

The set $P$ can contain $\Lambda$-nonlinear functionals.

It is clear that $K\subset L^{\prime}\cap P$. It follows that $K$ inherits two topologies: one from $L^{\prime}$ (its $\Lambda^{\prime}$-weak∗ topology, to which the conclusion of the theorem refers) and the other, $\tau_{1}$, from $P$. We will see that these two topologies coincide on $K$, and that $K$ is a closed subset of $P$.

We now prove that topologies $\tau_{1}$ and $\Lambda^{\prime}$-weak∗ coincide on $K$. Fix some $w^{\prime}_{0}\in K$. Then

[TABLE]

and

[TABLE]

are local bases in $(L^{\prime},\Lambda\text{-weak}^{*})$ and $(P,\tau_{1})$, respectively. From $K\subset L^{\prime}\cap P$ we have $K\cap W_{1}=K\cap W_{2}$, so topologies coincide on $K$.

Suppose $w^{\prime}_{0}$ is in the $\tau_{1}$-closure of $K$. If $S$ is from $C(\Lambda)$, then, from Lemma 3.1, operator $S(a\cdot)\colon\Lambda\rightarrow\Lambda$ belongs to $C(\Lambda)$ for all $a\in\Lambda$. For any $a,b\in\Lambda,$ $S\in C(\Lambda),$ $x,y\in L,$ $\varepsilon>0$, there is $\Lambda$-linear $w^{\prime}\in K$ from $V_{xa+yb,w^{\prime}_{0},S,\varepsilon}\cap V_{x,w^{\prime}_{0},S(a\cdot),\varepsilon}\cap V_{y,w^{\prime}_{0},S(b\cdot),\varepsilon}$ ($\tau_{1}$-neighborhood of $w^{\prime}_{0}$). Therefore, it holds

[TABLE]

[TABLE]

We have

[TABLE]

Since $\varepsilon>0$ was arbitrary, we see that

[TABLE]

for all $S\in C(\Lambda),$ i.e.

[TABLE]

for all $S\in C(\Lambda).$ For $S=L_{w^{\prime}_{0}(xa+yb)-w^{\prime}_{0}(x)a-w^{\prime}_{0}(y)b)}$ we have

[TABLE]

Hence $w^{\prime}_{0}(xa+yb)=w^{\prime}_{0}(x)a+w^{\prime}_{0}(y)b$ for all $x,y\in L,$ $a,b\in\Lambda$, so $w^{\prime}_{0}$ is $\Lambda$-linear.

Let $x\in B_{1}(L)$. For arbitrary $\varepsilon>0$ and $S\in C_{\Lambda}$ there is $w^{\prime}\in B_{1}(L^{\prime})$ such that $|\mathrm{sp}(S(w^{\prime}_{0}(x)))-\mathrm{sp}(S(w^{\prime}(x)))|<\varepsilon.$ Hence

[TABLE]

Next, from Theorem 2.2 we have

[TABLE]

for arbitrary $\varepsilon>0$. Thus $||w^{\prime}_{0}(x)||\leq 1$.

We have proven that $w^{\prime}_{0}\in B_{1}(L^{\prime})$, and that $B_{1}(L^{\prime})$ is a closed subset of $P$.

Since $P$ is $\tau_{1}$-compact, $B_{1}(L^{\prime})$ is a closed subset of $P$, and $\tau_{1}$ and $\Lambda^{\prime}$-weak∗ topology coincide on $B_{1}(L^{\prime})$, we have that $B_{1}(L^{\prime})$ is $\Lambda^{\prime}$-weak∗ compact. ∎

Corollary 3.3**.**

The unit ball $B_{1}(L)$ in $L$ is compact in $\Lambda$-weak∗ topology with the base

[TABLE]

for $x_{0},y_{j}\in L,$ $S_{j}\in C(\Lambda),$ $\delta>0.$

Proof.

For each $w\in L^{\prime}$ there is $y\in L$ such that $w^{\prime}(x)=[y,x]$ for all $x\in L$ (Theorem 2.1), so from Theorem 3.2 it follows that the unit ball $B_{1}(L)$ in $L$ is compact in given topology. ∎

Acknowledgement. The author was supported in part by the Ministry of education and science, Republic of Serbia, Grant 174034.