On Structure of Infinite B*-Matrices over Normed Fields

Abstract

This article is devoted to the investigation of infinite B*-matrices and linear operators over normed fields. Their structure is studied in the paper. Ideals and centers of the corresponding to them B*-algebras are scrutinized.

1. Introduction

Infinite matrices over the real field $\mathbf{R}$ and the complex field $\mathbf{C}$ and algebras of linear operators are indispensable in mathematics and its applications (see, for example [1,2,3,4] and references therein). In particular, C*-algebras play a very important role. However, for infinite matrices and analogs of such algebras over other normed fields, comparatively little is known. This is caused by their specific features and additional obstacles arising from the structure of fields [5,6,7,8,9,10,11,12].

A lot of results in the classical case are based on the fact that the real field R has a linear ordering compatible with its additive and multiplicative structure. Moreover, the complex field $\mathbf{C}$ is algebraically closed, norm complete, locally compact, and is the quadratic extension of $\mathbf{R}$. Besides $\mathbf{R}$ and $\mathbf{C}$, there are not any other commutative fields with archimedean multiplicative norms and complete relative to their norms.

Notice, that in the ultrametric (non-archimedean) case the algebraic closure of the field ${\mathbf{Q}}_p$ is not locally compact. There is not any ordering on the infinite normed field such as ${\mathbf{Q}}_{\mathbf{p}}$, ${\mathbf{C}}_{\mathbf{p}}$ or ${\mathbf{F}}_{\mathbf{p}}\left(t\right)$ compatible with its algebraic structure.

It is worth to mention that algebras over fields F other than $\mathbf{R}$ and $\mathbf{C}$, non-archimedean analysis, representations theory of groups and their applications develop fast in recent years [11,13,14,15,16,17,18]. Studies of matrices and linear algebras over fields with norms satisfying the strong triangle inequality are motivated not only by mathematical problems, but also by their applications in other sciences such as physics, quantum mechanics, quantum field theory, informatics, etc. (see, for example [19,20,21,22,23,24,25] and references therein).

This article is devoted to infinite B*-matrices over normed fields. Their structure is studied in the paper. Ideals and centers of B*-algebras are investigated in Theorems 1 and 2. Relations with operator theory and realizations of these algebras by algebras of infinite matrices are outlined. Theorems 3 and 4 about their embeddings into operator algebras are proven. This also provides tools for construction of wide families of such normed algebras. Realizations of elements of these algebras by infinite matrices are considered in Remark 3 and Example 8. Applications of obtained results are discussed in the conclusion.

All main results of this paper are obtained for the first time.

2. Embeddings of Normed*-Algebras

In this article, infinite matrices are considered over an infinite field F supplied with a multiplicative non-trivial norm denoted by ${|·|}_F$, where ${|·|}_F$ satisfies the strong triangle inequality:

$${|x+y|}_F\le {max\left(\right|x|}_F{,\left|y\right|}_F)$$

for each x and y in F. It is assumed that the field F is non-discrete and ${\mathsf{\Gamma }}_F:={\left\{\right|x|}_F:x\in F\setminus \left\{0\right\}\}\subset (0,\infty )=\{r\in \mathbf{R}:0<r<\infty \}$.

Henceforth, the field F is supposed to be complete relative to its norm, if some other will not be specified.

Matrices with matrix elements belonging to F are naturally related with linear operators in normed spaces over the field F. Suppose that $V=c_0(\alpha ,F)$ is a Banach space consisting of all vectors $x=(x_j:\forall j\in \alpha ,x_j\in F)$ subjected to the condition

$card\{j\in \alpha :|x_j|>ϵ\}<{\aleph }_0$ for each $ϵ>0$,

where V is supplied with the norm

$\left|x\right|={sup}_{j\in \alpha }\left|x_j\right|$,

where $\alpha$ is a set. For two normed spaces X and Y over the field F the linear space $L(X,Y)$ of all linear continuous operators $D:X\to Y$ is also normed:

$\left|D\right|:={sup}_{x\in X\setminus \left\{0\right\}}\left|Dx\right|/\left|x\right|$.

Let $X=c_0(\alpha ,F)$, $Y=c_0(\beta ,F)$, $D\in L(X,Y)$, where $\alpha$ and $\beta$ are sets. Then, to the operator D, a unique matrix $\left[D\right]=(d_{i,j}:i\in \beta ,j\in \alpha )$ corresponds such that $Dx=y$ and $y_i={\sum }_{j\in \alpha }{d}_{i,j}{x}_j$ with $d_{i,j}\in F$ for each $j\in \alpha$, $i\in \beta$, $x\in X$, $y\in Y$. The matrix $\left[D\right]$ is infinite, if $card\left(\alpha \right)\ge {\aleph }_0$ or $card\left(\beta \right)\ge {\aleph }_0$. Therefore, to any F-bimodule S contained in $L(X,Y)$, there corresponds a F-bimodule $\left[S\right]$ of matrices. In particular, $\left[S\right]$ is an algebra of matrices over F, if S is a subalgebra in $L(X,X)$.

Assume that A is a normed algebra over the field F such that a norm ${|·|}_A$ on A satisfies the following conditions:

${\left|a\right|}_A\in ({\mathsf{\Gamma }}_F\cup \left\{0\right\})$ for each $a\in A$, also

${\left|a\right|}_A=0$ if and only if $a=0$ in A,

${\left|ta\right|}_A={\left|t\right|}_F{\left|a\right|}_A$ for each $a\in A$ and $t\in F$,

${|a+b|}_A\le {max\left(\right|a|}_A{,\left|b\right|}_A)$ and

${\left|ab\right|}_A\le {\left|a\right|}_A{\left|b\right|}_A$ for each a and b in A.

Frequently, it is shortly written $|·|$ instead of ${|·|}_F$ or ${|·|}_A$.

We remind necessary definitions and notations.

Definition 1.

Assume that the field F is of the characteristic $char\left(F\right)\ne 2$. Assume also that $B_2=B_2\left(F\right)$ is the commutative associative algebra with one generator $i_1$ such that $i_1^2=-1$ and furnished with the involution ${\left(v{i}_1\right)}^{*}=-v{i}_1$ for each $v\in F$. Suppose that A is a subalgebra in $L(X,X)$ such that A is also a $B_2$-bimodule, where $X=c_0(\alpha ,F)$ is the Banach space over a field F, α is a set. Then, A is called a *-algebra if there is a continuous bijective (i.e., injective and surjective) F-linear operator

$\left(1\right)$$\mathcal{I}:A\to A$ such that

$\left(2\right)$$\mathcal{I}\left(ab\right)=\left(\mathcal{I}b\right)\left(\mathcal{I}a\right)$ and

$\left(3\right)$$\mathcal{I}\left(ga\right)=\left(\mathcal{I}a\right)g^{*}$ and $\mathcal{I}\left(ag\right)=g^{*}\left(\mathcal{I}a\right)$

$\left(4\right)$$\mathcal{I}\mathcal{I}a=a$

$\left(5\right)$$\left(\theta \right(y\left)\right)\left(ax\right)=\left(\theta \right(\left(\mathcal{I}a\right)y\left)\right)\left(x\right)$

for every a and b in A and $g\in B_2$ and x and y in X, where $\theta :X\hookrightarrow X^{\prime }$ is the canonical embedding of X into the topological dual space $X^{\prime }$ such that $\theta \left(y\right)x={\sum }_{j\in \alpha }{y}_j{x}_j$. For the sake of brevity, we can write $a^{*}$ instead of $\mathcal{I}a$. The mapping $\mathcal{I}$ is called the involution.

For two normed *-algebras A and B over F a map $\varphi :A\to B$ which is a continuous homomorphism of algebras and $\varphi \left(a^{*}\right)={\left(\varphi \left(a\right)\right)}^{*}$ for each $a\in A$ is called a *-homomorphism. If the *-homomorphism ϕ is bijective and ${\varphi }^{-1}:B\to A$ is also a *-homomorphism, then ϕ is called a *-isomorphism, and the normed *-algebras A and B are called *-isomorphic.

For the normed *-algebra A and $a\in A$ (or $U\subset A$) by $al{g}^{*}\left(a\right)$ (or $al{g}^{*}\left(U\right)$, respectively) will be denoted a minimal normed subalgebra in A containing a (or U, respectively). By $\overline{al{g}^{*}\left(U\right)}$ will be denoted the closure of $al{g}^{*}\left(U\right)$ in A.

Remark 1.

In Definition 1, $\theta \left(y\right)x$ is a particular case of a bilinear functional (see [11] and Remark 2 in more details).

Note that for matrices with entries in $\mathbf{C}$ corresponding to operators on $\mathbf{C}$-linear spaces, their block form with entries in $\mathbf{R}$ is frequently used, because each complex number can be written as the $2\times 2$ matrix with real entries. This is utilized for generating a complex *-algebra by the doubling procedure from a corresponding real algebra [1,3,5,26]. Using similar ideas, one can construct examples of Banach *-algebras over the field F other than $\mathbf{R}$ and $\mathbf{C}$ with $char\left(F\right)\ne 2$. This is evident for $F={\mathbf{Q}}_p$ with $\sqrt{-1}\notin {\mathbf{Q}}_p$ for primes p such that $p\ne 1\left(mod4\right)$ by Corollary 6 in Ch. 1, Section 4 in [25].

Example 1.

Let $X=c_0(\alpha ,F)$ be the Banach space such that $card\left(\alpha \right)\ge {\aleph }_0$. In view of Theorem 5.13 in [11], the direct sum $X\oplus X$ is isomorphic with X. In more details, one can take for the set α any fixed partition $\alpha ={\alpha }_1\cup {\alpha }_2$ with ${\alpha }_1\cap {\alpha }_2=\varnothing$ and $card\left({\alpha }_1\right)=card\left(\alpha \right)$, $card\left({\alpha }_2\right)=card\left(\alpha \right)$. Therefore, $c_0({\alpha }_j,F)=X_j$ is isomorphic with X for each $j\in \{1,2\}$ and $X_1\oplus X_2$ is isomorphic with X. This also implies that $L(X_j,X_j)$ is isomorphic with $L(X,X)$ as the Banach algebra for each $j\in \{1,2\}$.

For row vectors $x\in X$ and operators $a\in L(X,X)$, it is frequently written for convenience $xa$ instead of $x\left[a\right]$ or $a\left(x\right)$, where $\left[a\right]$ is a matrix corresponding to an operator a. Assume that $char\left(F\right)\ne 2$. For $char\left(F\right)\ne 2$ the algebra $B_2=B_2\left(F\right)$ can be embedded into $L(X,X)$ in the standard way, up to an automorphism of the Banach algebra $L(X,X)$, as induced by the formula $(x_1,x_2)i_1=(-x_2,x_1)$ for each $x=(x_1,x_2)\in X$, where $x_1\in X_1$ and $x_2\in X_2$, $i_1^2=-I$, I is the unit operator on X.

Example 2.

Let the condition of Example 1 be satisfied. Let B be a Banach subalgebra in $L(X,X)$ and let $\psi :B\to B$ be any continuous antiautomorphism of B. That is ψ is an F-linear map with continuous ψ and ${\psi }^{-1}$, ψ is a bijection (i.e., injection and surjection), $\psi \left(ab\right)=\psi \left(b\right)\psi \left(a\right)$ for each a and b in B. Such antiautomorphisms always exist, for example, $\psi \left(ab\right)=ba$ for each a and b in B [1,5,27]. Shortly, we denote $\psi \left(a\right)$ by $a^{\psi }$.

There are natural embeddings ${\eta }_j$ of $L(X_j,X_j)$ into $L(X,X)$ as the Banach algebras such that $(x_1,x_2){\eta }_1\left(a\right)=(x_{1}a,0)$ for each $a\in L(X_1,X_1)$, $(x_1,x_2){\eta }_2\left(b\right)=(0,x_{2}b)$ for each $b\in L(X_2,X_2)$, for each $x=(x_1,x_2)\in X$ with $x_1\in X_1$, $x_2\in X_2$, where $j\in \{1,2\}$. Then, to each $c\in L(X_1,X_2)$ or $d\in L(X_2,X_1)$, one can pose operators ${\eta }_3\left(c\right)$ with $(x_1,x_2){\eta }_3\left(c\right)=(0,x_{1}c)$ and ${\eta }_4\left(d\right)$ with $(x_1,x_2){\eta }_4\left(d\right)=(x_{2}d,0)$ for each $x_1\in X_1$ and $x_2\in X_2$. Shortly, ${\eta }_1\left(a\right)$, ${\eta }_2\left(b\right)$, ${\eta }_3\left(c\right)$, ${\eta }_4\left(d\right)$ will be denoted by $a,b,c,d$, respectively.

Let ψ be the antiautomorphism of $L(X_1,X_1)$. The Banach algebras $L(X_1,X_1)$ and $L(X_2,X_2)$ are isomorphic, hence ψ on $L(X_1,X_1)$ induces ψ on $L(X_2,X_2)$. For each $a_1\in L(X_1,X_1)$ let ${\eta }_5\left(a_1\right)\in L(X,X)$ be such that $(x_1,x_2){\eta }_5\left(a_1\right)=(x_1{a}_1,x_2{a}_1^{\psi })$ for each $x_1\in X_1$ and $x_2\in X_2$. Notice that, with $B_2=B_2\left(F\right)$ as in Example 1, evidently $A_{L(X_1,X_1)}:={\eta }_5\left(L(X_1,X_1)\right)+{\eta }_5\left(L(X_1,X_1)\right)i_1$ is the $B_2$-bimodule, where $U+W=\{x=u+w:u\in U,w\in W\}$ for subsets U and W in a F-linear space Y.

We take the antiautomorphism ψ of $L(X_1,X_1)$ and extend it on $A_{L(X_1,X_1)}$ such that $i_1^{\psi }=-i_1$ and ${(a_1+b_1{i}_1)}^{\psi }=a_1^{\psi }+i_1^{\psi }{b}_1^{\psi }$ for each $a_1$ and $b_1$ in ${\eta }_5\left(L(X_1,X_1)\right)$. For any $a_1$ and $b_1$ in $A_{L(X_1,X_1)}$, we put ${\left(a_1{i}_1\right)}^{*}=i_1^{*}{a}_1^{*}$ and ${(a_1+b_1{i}_1)}^{*}=a_1^{*}-b_1^{\psi }{i}_1$. Hence, $a_1^{*}=-i_1{a}_1^{\psi }{i}_1$, ${\left(a_1^{*}\right)}^{\psi }={\left(a_1^{\psi }\right)}^{*}$ and ${\left(a_1{b}_1{i}_1\right)}^{*}=-b_1^{\psi }{a}_1^{\psi }{i}_1$, since $b_1^{\psi }{a}_1^{\psi }={\left(a_1{b}_1\right)}^{\psi }$. Therefore, ${\left(\alpha \beta \right)}^{*}={\beta }^{*}{\alpha }^{*}$ for each $\alpha =a_1+b_1{i}_1$ and $\beta =a_2+b_2{i}_1$ with $a_1$, $a_2$, $b_1$, $b_2$ in ${\eta }_5\left(L(X_1,X_1)\right)$, since ${(a_1+a_2)}^{\psi }=a_1^{\psi }+a_2^{\psi }$, ${\left(a_1{a}_2\right)}^{\psi }=a_2^{\psi }{a}_1^{\psi }$, ${\left(a_1^{\psi }\right)}^{\psi }=a_1$, ${\left(a_1^{*}\right)}^{*}=a_1$. This implies that $A_{L(X_1,X_1)}$ is the closed subalgebra in $L(X,X)$ and $A_{L(X_1,X_1)}$ is supplied with the *-algebra structure.

Example 3.

Let the conditions of Example 2 be satisfied. We take any fixed antiautomorphism ψ on $L(X_1,X_1)$ extended on ${\eta }_5\left(L(X_1,X_1)\right)+{\eta }_5\left(L(X_1,X_1)\right)i_1$ and inducing the involution as in Example 2. Certainly, for any given subset V in $L(X_1,X_1)$, there exists a minimal closed subalgebra $A_V$ in $\left(L\right(X,X),|·\left|\right)$ such that ${\eta }_5\left(V\right)\subset A_V$, $A_V$ is the $B_2$-bimodule, $A_V^{*}=A_V$. This algebra $A_V$ is the intersection of all closed *-subalgebras $W_k$ in $\left(L\right(X,X),|·\left|\right)$ such that ${\eta }_5\left(V\right)\subset W_k$ and $W_k$ is the $B_2$-bimodule, where $W_k$ is with the involution inherited from ${\eta }_5\left(L(X_1,X_1)\right)+{\eta }_5\left(L(X_1,X_1)\right)i_1$ and $W_k\subset {\eta }_5\left(L(X_1,X_1)\right)+{\eta }_5\left(L(X_1,X_1)\right)i_1$. Evidently, $A_V$ is the closure in $\left(L\right(X,X),|·\left|\right)$ of a family ${\mathcal{F}}_V$ of all operators of the following form:

$a=f_1{\eta }_5\left(v_1\right)+...+f_n{\eta }_5\left(v_n\right)+f_{n+1}{\eta }_5\left(v_{n+1}\right)i_1+...+f_{n+m}{\eta }_5\left(v_{n+m}\right)$

belonging to $L(X,X)$ with $f_1,....,f_{n+m}$ in F, $v_1$, ..., $v_{n+m}$ in $V\cup V^{\psi }$, where $V^{\psi }=\{v^{\psi }:v\in V\}$.

Take $X_{1,1}=c_0({\alpha }_{1,1},F)$, $X_{1,2}=c_0({\alpha }_{1,2},F)$ with ${\alpha }_1={\alpha }_{1,1}\cup {\alpha }_{1,2}$, ${\alpha }_{1,1}\cap {\alpha }_{1,2}=\varnothing$, $card\left({\alpha }_{1,1}\right)=card\left({\alpha }_1\right)$, $card\left({\alpha }_{1,2}\right)=card\left({\alpha }_1\right)$, where $card\left({\alpha }_1\right)=card\left(\alpha \right)$, $card\left(\alpha \right)\ge {\aleph }_0$. Therefore, $X_1$ is isomorphic with $X_{1,1}\oplus X_{1,2}$ and with $X_{1,j}$ for each $j\in \{1,2\}$. In particular, V can be taken contained in $L(X_{1,1},X_{1,1})$, since there is the natural embedding ${\eta }_{1,1}:L(X_{1,1},X_{1,1})\hookrightarrow L(X_1,X_1)$ for $(X_{1,1},X_1)$ analogously to ${\eta }_1$ for $(X_1,X)$ described above. Therefore, there exists $V\ne IF$, $V\ne L(X_1,X_1)$ such that $A_V\ne A_{L(X_1,X_1)}$ and $A_V$ is the nontrivial *-algebra.

Example 4.

Let the conditions of Examples 1 and 2 be satisfied. For Banach spaces B and Y over F by $B{\widehat{\otimes }}_{F}Y$ is denoted the completion relative to the projective tensor product topology (see [11,28]) of the tensor product $B{\otimes }_{F}Y$ over the field F; or shortly $B\widehat{\otimes }Y$ instead of $B{\widehat{\otimes }}_{F}Y$, if F is specified. Let also $V\subset X_1\widehat{\otimes }{X}_1$, where $X_1=c_0({\alpha }_1,F)$. In view of Theorems 4.33, 4.40, and 4.41 in [11], $X_1\widehat{\otimes }{X}_1\hookrightarrow L_c(X_1,X_1)\hookrightarrow L(X_1,X_1)$ and $X_1\widehat{\otimes }{X}_1$ is the closed subalgebra in $L_c(X_1,X_1)$, where $L_c(X_1,X_1)$ denotes the algebra of all compact operators from $X_1$ into $X_1$. Notice that $L_c(X_1,X_1)$ is isomorphic with $X_1^{\prime }\widehat{\otimes }{X}_1$ by Theorem 4.41 [11], where $X_1^{\prime }$ is the topological dual space of $X_1$, $X_1^{\prime }=L(X_1,F)$. We say that $X_1\widehat{\otimes }{X}_1$ is the algebra of bicompact operators. In this case, it is possible to take $a^{\psi }=a^t$ for each $a\in X_1\widehat{\otimes }{X}_1$, where ${\left[a^t\right]}_{k,j}={\left[a\right]}_{j,k}$ for each j and k in ${\alpha }_1$. This implies that $A_V\subset {\eta }_5\left(X_1\widehat{\otimes }{X}_1\right)+{\eta }_5\left(X_1\widehat{\otimes }{X}_1\right)i_1\subset X\widehat{\otimes }X$ (see Example 3). Hence, $A_V\ne A_{L(X_1,X_1)}$.

Example 5.

Other examples of *-algebras which are proper subalgebras in $A_{L(X_1,X_1)}$ can be provided utilizing combinations of Examples 3 and 4. Note also that finite direct sums of *-algebras are *-algebras.

Assume that Λ is an infinite set. For Banach spaces $X_j$ over F for each $j\in \mathsf{\Lambda }$, let $c_0(X_j:j\in \mathsf{\Lambda })$ denote a Banach space over F such that each $x\in c_0(X_j:j\in \mathsf{\Lambda })$ has the form $x=(x_j:\forall j\in \mathsf{\Lambda },x_j\in X_j)$ and $\forall ϵ>0$, $card\{j:|x_j|>ϵ\}<{\aleph }_0$, $\left|x\right|={sup}_{j\in \mathsf{\Lambda }}\left|x_j\right|$, $fx=(f{x}_j:\forall j\in \mathsf{\Lambda },x_j\in X_j)$ for each $f\in F$, $x+y=(x_j+y_j:\forall j\in \mathsf{\Lambda },x_j\in X_j,y_j\in X_j)$ for each x and y in $c_0(X_j:j\in \mathsf{\Lambda })$. Assume also that $G_j$ is a Banach *-algebra over the field F for each $j\in \mathsf{\Lambda }$. Then, $G:=c_0(G_j:j\in \mathsf{\Lambda })$ is the Banach *-algebra with multiplication $xy=(x_j{y}_j:\forall j\in \mathsf{\Lambda },x_j\in G_j,y_j\in G_j)$ and inversion $x^{*}=(x_j^{*}:\forall j\in \mathsf{\Lambda },x_j\in G_j)$ for each x and y in G. We call G the $c_0$ direct sum of the Banach *-algebras $G_j$ and denote it also by $G={⨁}_{j\in \mathsf{\Lambda }}^{c_0}{G}_j$. Similarly, a $l_{\infty }$ direct sum can be defined.

3. Bilinear Functionals on Algebras

Definition 2.

For a topological algebra A over a field F and a subset S of A, the left annihilator of S is defined by $\mathsf{L}(A,S):=\{x\in A:xS=0\}$ and the right annihilator of S by $\mathsf{R}(A,S):=\{x\in A:Sx=0\}$. Shortly, they also will be denoted by $A_l\left(S\right):=\mathsf{L}(A,S)$ and $A_r\left(S\right):=\mathsf{R}(A,S)$ correspondingly.

The algebra A is called an annihilator algebra if it satisfies conditions $\left(6\right)$–$\left(8\right)$:

$\left(6\right)$$A_l\left(A\right)=A_r\left(A\right)=0$ and

$\left(7\right)$$A_l\left(J_r\right)\ne 0$ and

$\left(8\right)$$A_r\left(J_l\right)\ne 0$

for all proper closed right $J_r$ and left $J_l$ ideals in A.

Definition 3.

Let A and B be two Banach algebras over the normed field F. Let $A{\widehat{\otimes }}_{F}B$ be the completion relative to the projective tensor product topology (see [11,28]) of the tensor product $A{\otimes }_{F}B$ over the field F.

Suppose that B is a Banach algebra over the normed field F, and x is an element in B. It will be said that x has a left core quasi-inverse y if for any complete normed (valued) field extension H of F an element $y\in B_H$ exists satisfying the equality $x+y+yx=0$, where $B_H=B{\widehat{\otimes }}_{F}H$, where H is such that ${\left|b\right|}_H={\left|b\right|}_F$ for each $b\in F$. In particular, if only the field $H=F$ is considered, it is called a left quasi-inverse.

Assume that A is a unital Banach algebra over F. Suppose also that an element $x\in A$ has the property: for any complete normed (valued) field extension G of F the left inverse ${(1+yx)}_l^{-1}$ exists in $A_G$ for each $y\in A_G$, where G is normed such that ${\left|b\right|}_G={\left|b\right|}_F$ for each $b\in F$. Then, we call x a generalized core nil-degree element. The family of all generalized core nil-degree elements of A is called a core radical and it is denoted by $R_c\left(A\right)$. A radical of the algebra A is denoted by $R\left(A\right)$.

Remark 2.

We recall known necessary facts about bilinear functionals. Let E and Y be Banach spaces over the field F. A space $Bil(E,Y;F)$ of all continuous bilinear functionals $T:E\times Y\to F$ is normed and isomorphic with $L(E,L(Y,F\left)\right)$ and with $L(Y,L(E,F\left)\right)$ (Ch. 4 in [11]).

Let $X=c_0(\mathbf{N},F)$ be a Banach space over the field F (F is complete relative to its multiplicative norm, as it was supposed above). Let $S:X\to X$ be a compact operator. Then, for each x and y in $L(X,X)$, the trace $Tr\left(xSy\right)$ exists and

$\left(9\right)$$\left|Tr\right(xSy\left)\right|\le \left|x\right|\left|S\right|\left|y\right|$.

Indeed, let S be a marked compact operator, $S\in L_c(X,X)$. A series ${\sum }_{j=1}^{\infty }{b}_j$ with $b_j\in F$ for each $j\in \mathbf{N}$ converges in F if and only if ${lim}_{j\to \infty }{b}_j=0$ by Proposition 23.1 in [16], since the field F is complete relative to its norm. Notice that the trace $Tr\left(C\right)={\sum }_j{C}_{j,j}$ is defined for each compact operator $C\in L_c(X,X)$ by Theorem 4.40 in [11] (see also Definition 2.2 in [9]). By virtue of Theorems 4.37 and 4.40 in [11] $L_c(X,X)$ is the closed two-sided ideal in $L(X,X)$. This implies that $Tr\left(xSy\right)$ exists for each x, y in $L(X,X)$ (see also Theorem 2 in [9]). Since $\left|Tr\right(C\left)\right|\le \left|C\right|$ and $\left|xSy\right|\le \left|x\right|\left|S\right|\left|y\right|$, then $\left|Tr\right(xSy\left)\right|\le \left|x\right|\left|S\right|\left|y\right|$ for each x, y in $L(X,X)$.

Notice that for a subalgebra A in the algebra $L(X,X)$ the trace $Tr\left(xSy\right)$ exists for each x, y in A as the restriction of this bilinear continuous functional from $L(X,X)$ on A. In this article bilinear functionals are considered on Banach spaces.

Lemma 1.

Let G be a normed *-algebra over F of $char\left(F\right)\ne 2$ such that $G\subset L(X,X)$ and $G=G_1\oplus G_1{i}_1$ with $G_1$ being a subalgebra in $L(X_1,X_1)$ (see Example 3). Let also ${(·,·)}_1:G_1^2\to F$ be a bilinear functional on $G_1$ such that for each nonzero $a_1\in G_1\setminus \left\{0\right\}$ there exists $a_2\in G_1$ with ${(a_1,a_2)}_1\ne 0$. Then, the bilinear functional ${(·,·)}_1$ has a bilinear extension $(·,·):G^2\to F$ such that for each nonzero $a\in G\setminus \left\{0\right\}$ there exists $b\in G$ with $(a,b)\ne 0$.

Proof. 

We put

$\left(10\right)$$(a,b)={(a_1,a_2)}_1-{(b_1,b_2)}_1$

for each $a=a_1+b_1{i}_1$ and $b=a_2+b_2{i}_2$ in G with $a_1$, $a_2$, $b_1$ and $b_2$ in $G_1$. If $a\ne 0$, then either $a_1\ne 0$ or $b_1\ne 0$. For $a_1\ne 0$ one can take $a_2$ in $G_1$ with ${(a_1,a_2)}_1\ne 0$ and put $b_1=0$. For $b_1\ne 0$ one can take $b_2\in G_1$ with ${(b_1,b_2)}_1\ne 0$ and put $a_2=0$. □

4. ${\mathbf{B}}^{*}$-Matrices and Algebras

Definition 4.

Let A be an normed algebra over the field F, (see Introduction), satisfying the following conditions:

$\left(11\right)$ A is a Banach *-algebra and

$\left(12\right)$ there exists a bilinear functional $(·,·):A^2\to F$ such that $\left|\right(x,y\left)\right|\le q\left|x\right|\left|y\right|$ for all x and y in A, where $0<q<\infty$ is a constant independent of x and y,

$\left(13\right)$$(x,y)=(y,x)$ and $(x,y)=(x^{*},y^{*})$ for each x and y in A,

$\left(14\right)$ if $(x,y)=0$ for each $y\in A$, then $x=0$;

$\left(15\right)$$(xy,z)=(x,z{y}^{*})$ for every x, y and z in A,

$\left(16\right)$$x{x}^{*}\ne 0$ for each nonzero element $x\in A\setminus \left(0\right)$.

Then we call A a $B^{*}$-algebra. If an operator D belongs to the $B^{*}$-algebra A, $A\subset L(X,X)$, $X=c_0(\alpha ,F)$, then the corresponding matrix $\left[D\right]$ is called a $B^{*}$-matrix.

Lemma 2.

For a *-subalgebra A of $L(X,X)$ with $X=c_0(\mathbf{N},F)$, a bilinear functional $(·,·)$ satisfying conditions $\left(12\right)$, $\left(13\right)$, and $\left(15\right)$ exists.

Proof. 

We put $(x,y)=Tr\left(x^{*}Sy\right)$, where S is a marked compact operator such that $S^{*}=S$, $S\in L_c(X,X)$, $X=c_0(\mathbf{N},F)$ (see Remark 2). From the inequality $\left(9\right)$ it follows that Condition $\left(12\right)$ is valid. From $Tr\left(C^{*}\right)={\left(Tr\left(C\right)\right)}^{*}=Tr\left(C\right)$ for each $C\in L_c(X,X)$ and ${\left(x^{*}Sy\right)}^{*}=y^{*}Sx$ property $\left(13\right)$ follows, since $b^{*}=b$ for each $b\in F$. Then, using the identity $Tr\left(CD\right)=Tr\left(DC\right)={\sum }_{k,j}{C}_{k,j}{D}_{j,k}$ for each $C\in L_c(X,X)$ and $D\in L(X,X)$ we deduce that $(xy,z)=Tr\left(y^{*}{x}^{*}Sz\right)=Tr\left(x^{*}Sz{y}^{*}\right)=(x,z{y}^{*})$ for every x, y and z in A, since ${\left(xy\right)}^{*}=y^{*}{x}^{*}$. □

Lemma 3.

Let $X=c_0(\mathbf{N},F)$ and let A be a Banach *-algebra over F such that $A\subset L(X,X)$. Let also either $\left(1\right)$ or $\left(2\right)$ be satisfied:

(i)

If $char\left(F\right)\ne 2$ and for each $a\in A\setminus \left\{0\right\}$ there exists a normed extension $K_a$ of the field F such that there exists a *-homomorphism ${\varphi }_a$ from $K_a{\widehat{\otimes }}_F\overline{al{g}^{*}\left(a\right)}$ into a *-algebra $B_a$, such that $B_a\subset L(X_{K_a},X_{K_a})$ with $X_{K_a}=c_0(\mathbf{N},K_a)$ and ${\varphi }_a\left(a{a}^{*}\right)=J^{-1}DJ$, where $J\in L(X_{K_a},X_{K_a})$ is an invertible operator and a matrix $\left[D\right]$ of an operator $D\in L(X_{K_a},X_{K_a})$ is diagonal and nonzero, $\left[D\right]\ne 0$; or

(ii)

if $char\left(F\right)=0$ and for each $a\in A\setminus \left\{0\right\}$ there exists a *-homomorphism ${\psi }_a:al{g}^{*}\left(a\right)\to A_{L(X_1,X_1)}$ with nonzero image ${\psi }_a\left(a\right)$, ${\psi }_a\left(a\right)\ne 0$ (see Examples 2 and 3),

then conditions $\left(12\right)$–$\left(16\right)$ are also valid. Moreover,

(iii)

if $A_j$ is a $B^{*}$-algebra over F for each $j\in \mathbf{N}$,

then $A:={⨁}_{j\in \mathbf{N}}^{c_0}{A}_j$ and $B:={⨁}_{j\in \mathbf{N}}^{l_{\infty }}{A}_j$ are $B^{*}$-algebras.

Proof. 

In cases $\left(i\right)$ and $\left(ii\right)$, in view of Lemmas 2, 1, and Formula $\left(10\right)$, Conditions $\left(12\right)$, $\left(13\right)$ and $\left(15\right)$ are satisfied. Indeed, using injective *-homomorphism it is possible to choose $S\in L_c(X,X)$ for which the decomposition $S=T^{-1}YT$ is such that $T:X\to X$ is an automorphism of the Banach space X and $S^{*}=S$, also $Y{e}_j=Y_{j,j}{e}_j$ with $Y_{j,j}\ne 0$ for each j, while $Y_{i,j}=0$ for each $i\ne j$, where $\{e_k:k\in \mathbf{N}\}$ is the standard basis of X, since $f^{*}=f$ for each $f\in F$. Then, we get property $\left(14\right)$, since $Tr\left(x^{*}Sy\right)\in F$.

In case $\left(i\right)$, we deduce that ${\varphi }_a\left(a\right){\left({\varphi }_a\left(a\right)\right)}^{*}\ne 0$, hence $a{a}^{*}\ne 0$, since ${\varphi }_a$ is the *-homomorphism and ${\varphi }_a\left(a{a}^{*}\right)={\varphi }_a\left(a\right){\left(\varphi \left(a\right)\right)}^{*}$. This implies $\left(16\right)$.

In case $\left(ii\right)$, let $x={\psi }_a\left(a\right)$. By the imposed conditions in $\left(2\right)$ x is nonzero, $x\ne 0$. On the other hand, $\left(bx\right){\left(bx\right)}^{*}=b\left(x{x}^{*}\right)b^{*}$ and ${\left(x{x}^{*}\right)}^{*}=x{x}^{*}$ for each $b\in A_{L(X_1,X_1)}$. Let $E_{j,k}={e^{\prime }}_j\otimes e_k$, ${e^{\prime }}_j=\theta \left(e_j\right)$ for each j and k in ${\alpha }_1$ (see also Definition 1 and Examples 2, 3). Therefore, considering $b\in A$ of the form $b=b_1+b_2{i}_1$ with $b_l={\sum }_{k,j}{f}_{l;j,k}{E}_{j,k}$ with $f_{l;j,k}\in F$ for every $l\in \{1,2\}$, j and k in $\mathbf{N}$, one finds coefficients $f_{l;j,k}$ such that $\left(bx\right){\left(bx\right)}^{*}\ne 0$, since $E_{j,k}\in X_1{{\displaystyle \widehat{⨂}}}_F{X}_1\hookrightarrow L(X_1,X_1)$ for each $j,k$ in ${\alpha }_1$, $x\ne 0$. Note that $\left(bx\right){\left(bx\right)}^{*}\ne 0$ implies that $x{x}^{*}\ne 0$ and consequently, $a{a}^{*}\ne 0$, since the algebra A is associative and ${\psi }_a$ is the *-homomorphism. Thus, property $\left(16\right)$ also is fulfilled.

$\left(iii\right)$. For each $x\in A$ (or $x\in B$), there is the decomposition

$\left(17\right)$$x={⨁}_{j\in \mathbf{N}}^{c_0}{x}_j$ (or $x={⨁}_{j\in \mathbf{N}}^{l_{\infty }}{x}_j$, respectively) with $x_j\in A_j$ for each $j\in \mathbf{N}$. Therefore, $x{x}^{*}={⨁}_{j\in \mathbf{N}}^{c_0}{x}_j{x}_j^{*}$ (or $x{x}^{*}={⨁}_{j\in \mathbf{N}}^{l_{\infty }}{x}_j{x}_j^{*}$, respectively). Hence if $x\ne 0$, then $x{x}^{*}\ne 0$.

For each $j\in \mathbf{N}$ there exists a constant $w_j>0$ such that $|{(x_j,y_j)}_j|\le w_j|x_j\left|\right|y_j|$ for each $x_j$, $y_j$ in $A_j$, where ${(x_j,y_j)}_j$ denotes the bilinear functional on $A_j$ satisfying the conditions of Definition 4. We choose $\pi \in F$ such that $0<\left|\pi \right|<1$, because the field F is infinite non-discrete. For each $j\in \mathbf{N}$ there exists $l\left(j\right)\in \mathbf{N}$ such that ${\left|\pi \right|}^j<w_j{\left|\pi \right|}^{l\left(j\right)}\le {\left|\pi \right|}^{j-1}$. Let

$\left(18\right)$$(x,y)={\sum }_{j=1}^{\infty }{(x_j,y_j)}_j{\pi }^{l\left(j\right)}$,

then $\left|\right(x,y\left)\right|\le \left|x\right|\left|y\right|/(1-|\pi \left|\right)$ for each x and y in A (or B, respectively). This implies that the bilinear functional given by Formula $\left(18\right)$ satisfies conditions of Definition 4. □

Lemma 4.

If $J_r$ and $J_l$ are proper or improper right and left ideals in a $B^{*}$-algebra A, then $\mathsf{L}(A,J_r)$ and $\mathsf{R}(A,J_l)$ are orthogonal relative to the family of bilinear functionals $\{{(·,·)}_a:a\in A\}$ complements of the sets $J_r^{*}$ and $J_l^{*}$ in the Banach space A, where ${(x,y)}_a=(ax,ay)$ for every a, x and y in A.

Proof. 

If $x\in \mathsf{L}(A,J_r)$ (see Definition 2), then $x{J}_r=\left(0\right)$, hence $(ax{J}_r,aA)=0$ for each $a\in A$ and consequently, $(ax,aA{J}_r^{*})=0$ by identity $\left(15\right)$ and inevitably $(ax,a{J}_r^{*})=0$. This means that $x\in A\ominus J_r^{*}$ relative to $\{{(·,·)}_a:a\in A\}$, that is $\mathsf{L}(A,J_r)$ is the orthogonal complement of $J_r^{*}$. Similarly, $\mathsf{R}(A,J_l)$ is the orthogonal complement of $J_l^{*}$ in A as the Banach space relative to the family $\{{(·,·)}_a:a\in A\}$ of bilinear functionals. □

Proposition 1.

Any $B^{*}$-algebra A is dual.

Proof. 

If $J_r$ and $J_l$ are right and left ideals in A, then by Lemma 4 $\mathsf{R}(A,\mathsf{L}(A,J_r))=\mathsf{R}(A,A\ominus J_r^{*})=A\ominus (A\ominus J_r)=J_r$ and analogously $\mathsf{L}(A,\mathsf{R}(A,J_l))=J_l$, since $A^{*}=A$ and ${\left(J_r^{*}\right)}^{*}=J_r$. □

Theorem 1.

Any $B^{*}$-algebra A over the spherically complete field F with $R_c\left(A\right)=R\left(A\right)$ is representable as the direct sum of its two-sided minimal closed ideals, which are simple $B^{*}$-algebras and pairwise orthogonal relative to the family of bilinear functionals $\{{(·,·)}_a:a\in A\}$.

Proof. 

By virtue of Theorem 8 in [9] and Proposition 1, the algebra A is the completion (relative to the norm) of the direct sum of its minimal closed two-sided ideals which are simple dual subalgebras (see also Definition 3). Consider a two-sided minimal closed non null ideal J in A. The involution mapping $x\mapsto \mathcal{I}x=x^{*}$ provides from it the minimal closed two-sided ideal $J^{*}$ due to Condition $\left(1\right)$.

Suppose that $J^{*}\ne J$, then $J{J}^{*}=\left(0\right)$, since the ideal J is minimal. From $aJ\subset J$ and $Ja\subset J$ for each $a\in A$ we deduce that $AJ{J}^{*}A=\left(0\right)$. Together with condition $\left(16\right)$ imposed on the $B^{*}$-algebra, this would imply that $x=0$ for each $x\in J$ contradicting $J\ne \left(0\right)$. Thus, $J^{*}=J$.

Notice that properties $\left(11\right)$–$\left(13\right)$ and $\left(15\right)$ for J are inherited from that of A. Then, condition $\left(16\right)$ on A implies that $J^2\ne \left(0\right)$, since $J^{*}=J$ and $AJ\subseteq J$, also $JA\subseteq J$. However, J is minimal, hence $J^2=J$. Therefore, property $\left(14\right)$ on J follows from that of on A and $\left(15\right)$ and $J^2=J$, since for each $u\in J$ there exists x and y in J with $u=xy$ and $(u,z)=(xy,z)=(x,z{y}^{*})$ for all $z\in A$, also since $z{y}^{*}\in J$. Then, for each $y\in J\setminus \left(0\right)$ an element $x\in J\setminus \left(0\right)$ exists such that $xy\ne 0$, hence $u=xy\in J\setminus \left(0\right)$. Then, we have that $u{u}^{*}\ne 0$ by $\left(16\right)$ on A. Hence, $\left(xy\right){\left(xy\right)}^{*}\ne 0$, consequently, $y{y}^{*}\ne 0$, since the algebra A is associative and $x\left(y{y}^{*}\right)x^{*}\ne 0$. Therefore, property $\left(16\right)$ on J is valid. Thus, J is the $B^{*}$-algebra.

If J and S are two distinct minimal closed two-sided ideals in A, then $JS=\left(0\right)$. From Lemma 4, it follows that $S\subset \mathsf{R}(A,J)=A\ominus J^{*}=A\ominus J$. Thus, these ideals J and S are orthogonal relative to the family $\{{(·,·)}_a:a\in A\}$ of bilinear functionals.

Using condition $\left(14\right)$ and Lemma 4, we infer that A is the direct sum of its two-sided minimal closed ideals. □

Theorem 2.

Let A be a simple unital $B^{*}$-algebra over the spherically complete field F with $R_c\left(A\right)=R\left(A\right)$ and let a division algebra G be provided by Theorem 2 in [10]. Then, the following conditions are equivalent:

(i)

$A_G$ is finite dimensional over G;

(ii)

$A_G$ is unital;

(iii)

the center $Z\left(A_G\right)$ of $A_G$ is non-null.

Proof. 

Let $\{w_j:j\in \mathsf{\Lambda }\}$ be a maximal system of irreducible idempotents provided by Theorem 2 in [10].

$\left(i\right)\Rightarrow \left(ii\right)$. If $A_G$ is finite dimensional over G, then according to Theorem 1, a maximal system $\{w_j:j\in \mathsf{\Lambda }\}$ of irreducible idempotents is finite, that is $\mathit{card}\left(\mathsf{\Lambda }\right)<{\aleph }_0$. Then, their sum $w={\sum }_{j\in \mathsf{\Lambda }}{w}_j$ is the idempotent fulfilling the condition $x={\sum }_{j\in \mathsf{\Lambda }}x{w}_j=xw$ and $x={\sum }_{j\in \mathsf{\Lambda }}{w}_{j}x=wx$. Thus, w is the unit in $A_G$.

$\left(ii\right)\Rightarrow \left(iii\right)$. If $A_G$ contains a unit w, then $Z\left(A_G\right)$ contains w, consequently, $Z\left(A_G\right)$ is non-null.

$\left(iii\right)\Rightarrow \left(i\right)$. Let $Z\left(A_G\right)\ne \left(0\right)$ and x be a non-zero element of $Z\left(A_G\right)$, $x\ne 0$. In view of Theorem 2 in [10] $x{w}_j=\left(x{w}_j\right)w_j=w_{j}x{w}_j=w_j^{2}x{w}_j$, hence $x{w}_j=b_j{w}_j=w_j{b}_j{w}_j$, where $b_j\in G$. Thus $\left(b_j{w}_j\right)w_j=w_j\left(b_j{w}_j\right)$. Therefore, $x={\sum }_{j}x{w}_j={\sum }_j{b}_j{w}_j$ and hence $b_j{w}_{j,k}=b_j{w}_j{w}_{j,k}=x{w}_j{w}_{j,k}=x{w}_{j,k}=w_{j,k}x=w_{j,k}{w}_{k}x=w_{j,k}x{w}_k=w_{j,k}{b}_k{w}_k$. Similarly, $b_k{w}_{k,j}=w_{k,j}{b}_j{w}_j$, consequently, $b_j{w}_{j,k}{w}_{k,j}=b_j{w}_j=w_{j,k}{b}_k{w}_k{w}_{k,j}=w_{j,k}{b}_k{w}_{k,j}$ and hence ${\sum }_j{b}_j{w}_j=b_k{w}_k+{\sum }_{j,j\ne k}{w}_{j,k}{b}_k{w}_{k,j}={\sum }_j{w}_{j,k}{b}_k{w}_{k,j}$.

Note that $w_j{A}_G{w}_j=G{w}_j$ for each j, where $w_j$ plays the role of the unit in $G{w}_j$. Then,

$G{w}_j\supseteq w_j\left(w_{j,k}{A}_G{w}_{k,j}\right)w_j=w_{j,k}{A}_G{w}_{k,j}$

$=w_{j,k}\left(w_k{A}_G{w}_k\right)w_{k,j}\supseteq w_{j,k}\left(w_{k,j}{A}_G{w}_{j,k}\right)w_{k,j}=w_j{A}_G{w}_j=G{w}_j$ for each j and k, hence $G{w}_k\ni b\mapsto w_{j,k}b{w}_{k,j}\in G{w}_j$ is the isomorphism of normed algebras $G{w}_j$ with $G{w}_k$ for each j and k.

Therefore, the sum ${\sum }_j{w}_{j,k}{b}_k{w}_{k,j}={\sum }_j{w}_{j,k}{w}_k{b}_k{w}_k{w}_{k,j}$ may converge only if it is finite. Thus, the algebra $A_G$ is finite dimensional over G. □

Remark 3.

For a Banach space H over the field F and a set α by $c_0(\alpha ,H)$ is denoted a $c_0$ direct sum of α copies of H such that $c_0(\alpha ,H)$ is a Banach space consisting of all vectors $y=(y_j\in H:j\in \alpha )$ with $\left|y\right|={sup}_{j\in \alpha }\left|y_j\right|<\infty$ and such that for each $t>0$ a set $\{j\in \alpha :|y_j|>t\}$ is finite. In particular, for the Banach space $X=c_0(\alpha ,F)$ over a spherically complete field F, there exists a topological dual space $X^{\prime }$ of all continuous F-linear functionals $h:X\to F$ (see Ch. 2 and 5 in [11] or Ch. 8 in [29]). Each vector x in $X_H$ has the following decomposition: $x={\sum }_{j\in \alpha }{e}_j{x}_j$, where $x_j\in H$, $e_j\in X_H$ with $e_j=({\delta }_{i,j}:i\in \alpha )$, ${\delta }_{i,j}$ denotes the Kronecker delta symbol such that ${\delta }_{i,j}=0$ for each $i\ne j$ in α, ${\delta }_{j,j}=1$ for each $j\in \alpha$.

Then, for a division algebra H over the spherically complete field F and a Banach H-bimodule $X_H=c_0(\alpha ,H)$ we consider a bounded F-linear right H-linear operator C from $X_H$ into $X_H$, that is $C\left(xb\right)=\left(Cx\right)b$ for each $x\in X_H$ and $b\in H$. The embedding of F into H as $F{1}_H$, where $1_H$ is a unit element in H, induces a F-linear embedding of X into $X_H$. In this case to each $x\in X$ there corresponds a continuous F-linear right H-linear functional $x^{\prime }=\theta \left(x\right)$ such that $\theta \left(x\right)y={\sum }_{j\in \alpha }{x}_j{y}_j$ for each $x\in X$ and $y\in X_H$. This induces a natural embedding $\theta :X\hookrightarrow L_r(X_H,H)$, where $L_r(X_H,H)$ denotes a space of all bounded F-linear right H-linear operators from $X_H$ into H (see Ch. 3 and 5 in [11], Proposition 23.1 in [16]). Therefore, for the operator C and for each i and j in α, there exists a matrix element $\theta \left(e_j\right)C{e}_i=:C_{j,i}$. Then, by $L_{r,d}(X_H,X_H)$ is denoted the space of all bounded F-linear right H-linear operators C from $X_H$ into $X_H$ satisfying the condition:

(i)

for each $t>0$ a finite subset γ in a set α exists such that $|C_{j,k}|<t$ for each j and k with either $j\in \alpha \setminus \gamma$ or $k\in \alpha \setminus \gamma$.

Theorem 3.

Let A be a spherically complete simple unital $B^{*}$-algebra over a spherically complete field F with $R_c\left(A\right)=R\left(A\right)$. Let also G be a division algebra provided by Theorem 2 in [10] such that $s^{1/2}\in G$ for each $s\in G$, also $G\subset A$ and $G^{*}=G$. Then a Banach G-bimodule $X_G$ exist such that A and $L_{r,d}(X_G,X_G)$ are isomorphic as the Banach right G-modules and as F-algebras.

Proof. 

By the conditions of this theorem, a division algebra G is such that $wAw\subset Gw$ for each irreducible idempotent w in A. Put $H=G\cap G^{*}$. From $G=G^{*}$, it follows that $H=G$. If $b\in H$, then $b^{1/2}\in G$ and ${\left(b^{1/2}\right)}^{*}={\left(b^{*}\right)}^{1/2}\in G$, since $H^{*}=H$, consequently, $b^{1/2}\in H$.

For each irreducible idempotent w such that $w{A}_{G}w=G$ (see the proof of Theorem 3 in [10]) one gets that $w{w}^{*}\ne 0$, since A is the $B^{*}$-algebra over F. Then, $\left(w{w}^{*}\right){\left(w{w}^{*}\right)}^{*}\ne 0$, hence $w{w}^{*}w{w}^{*}\ne 0$ and consequently, $w^{*}w{w}^{*}\ne 0$ implying that $w{w}^{*}w\ne 0$, since ${\left(w^{*}w{w}^{*}\right)}^{*}=w{w}^{*}w$ and $c^{**}=c$ for each $c\in A$. Therefore, $w^{*}w\ne 0$ also.

Since w is the irreducible idempotent and $A^{*}=A$, then $w^{*}$ is the irreducible idempotent in the $B^{*}$-algebra A. Then, we deduce that $w^{*}w{w}^{*}\in \left(w^{*}{A}_G{w}^{*}\right)w^{*}\subseteq G^{*}{w}^{*}={\left(wG\right)}^{*}$, since $A^{*}=A$, consequently, an element $s\in G^{*}\setminus \left(0\right)$ exists such that $w^{*}w{w}^{*}=s{w}^{*}$, since $w^{*}w{w}^{*}\ne 0$. The latter implies $w^{*}w{w}^{*}w=s{w}^{*}w$. However, the elements $w^{*}w{w}^{*}w$ and $w^{*}w$ are self-adjoint, hence $s{w}^{*}w=w^{*}w{s}^{*}$ and consequently,

$w^{*}w{\left(s^{*}\right)}^{-1}=s^{-1}{w}^{*}w$.

We put $v=s^{-1}{w}^{*}w$, hence

$v^{*}=w^{*}w{\left(s^{*}\right)}^{-1}=s^{-1}{w}^{*}w=v$ and

$v^2=s^{-1}{w}^{*}w{s}^{-1}{w}^{*}w=s^{-1}{w}^{*}w{w}^{*}w{\left(s^{*}\right)}^{-1}$

$=\left(s^{-1}\left(s{w}^{*}w\right)\right){\left(s^{*}\right)}^{-1}=w^{*}w{\left(s^{*}\right)}^{-1}$

$=s^{-1}{w}^{*}w=v$.

Thus, v is the self-adjoint idempotent. On the other hand, $A_{G}v=A_G{s}^{-1}{w}^{*}w\subseteq A_{G}w$ and $A_{G}v\ne 0$ and the idempotent w is irreducible, hence the idempotent v is also irreducible, since $A_{G}w$ is the non-null minimal left ideal in $A_G$.

Then, from the proof of Theorem 2 in [10] it follows that ${\left(v{A}_{G}v\right)}^{*}=v^{*}{A}_G^{*}{v}^{*}=v{A}_{G}v$ is the self-adjoint division algebra for each such irreducible self-adjoint idempotent v, consequently, $v{A}_{G}v\subseteq Hv$. By the conditions of this theorem we have $A=A_G$.

The algebra A is simple, that is by the definition each its two-sided ideal coincides with either $\left(0\right)$ or A.

Next we take a maximal orthogonal system $\{w_j:j\in \mathsf{\Lambda }\}$ of self-adjoint idempotents in A and for them elements $w_{j,k}$ as in Theorem 2 in [10], where $\mathsf{\Lambda }$ is a set. Hence, $w_{j,k}{w}_{j,k}^{*}\in w_{j}A{w}_j$ and $b=b_{j,k}\in H$ exists such that $w_{j,k}{w}_{j,k}^{*}=b{w}_j$. Then, $b{w}_j=w_j{b}^{*}$, since $w_j^{*}=w_j$ and ${\left(w_{j,k}{w}_{j,k}^{*}\right)}^{*}=w_{j,k}{w}_{j,k}^{*}$. Moreover, $b\ne 0$, since $w_{j,k}$ is non null and hence $w_{j,k}{w}_{j,k}^{*}$ is non-null. For $v_{j,k}={\left(b_{j,k}\right)}^{-1/2}{w}_{j,k}$, we deduce that $v_{j,k}{v}_{j,k}^{*}=w_j$, since

$b^{-1/2}{w}_{j,k}{w}_{j,k}^{*}{\left(b^{-1/2}\right)}^{*}=b^{-1/2}b{w}_j{\left(b^{-1/2}\right)}^{*}$

$=w_j{\left(b^{1/2}\right)}^{*}{\left(b^{-1/2}\right)}^{*}=w_j{\left(b^{-1/2}{b}^{1/2}\right)}^{*}=w_j$,

since A is associative and $b^{-1/2}\in H$ for each non null b in H, where $b=b_{j,k}$.

Thus, it is possible to choose an element $w_{j,k}$ such that $w_{j,k}{w}_{j,k}^{*}=w_j$ for each k. Taking a marked element $j=j_0$ and setting $w_{k,j}=w_{j,k}^{*}$ and $w_{l,k}=w_{l,j}{w}_{j,k}$ for each l and k one gets $w_{l,k}^{*}=w_{j,k}^{*}{w}_{l,j}^{*}=w_{k,j}{w}_{j,l}=w_{k,l}$ and $w_{k,k}=w_k$, also $w_{k,l}{w}_{i,h}={\delta }_{l,i}{w}_{k,h}$ for every $h,i,k,l$. Thus, elements $w_{l,k}$ can be chosen such that $w_{l,k}^{*}=w_{k,l}$ for each l and k.

If the statement of this theorem for the spherical completion $\tilde{H}$ of H is proven, then it will imply the statement of this theorem for H. So the case of the spherically complete division algebra H is sufficient. Then, A and H considered as the Banach spaces over the spherically complete field F are isomorphic with $c_0(\alpha ,F)$ and H with $c_0(\beta ,F)$ due to Theorems 5.13 and 5.16 in [11], where $\beta \subset \alpha$.

From the proof of Theorem 3 in [10], it follows that the sum $B:={\sum }_{j,k}{w}_{j}A{w}_k$ is dense in A. Conditions $\left(12\right)$, $\left(13\right)$, $\left(15\right)$ imply that $(xy,z)=(y,x^{*}z)$, since $t^{*}=t$ for each $t\in F$. Therefore, from properties $\left(12\right)$, $\left(13\right)$, $\left(15\right)$ it follows that if $j\ne h$ or $k\ne l$, then $(w_{j}x{w}_k,w_{h}z{w}_l)=0$ for each x and z in A, since

$(w_{j}x{w}_k,w_{h}z{w}_l)=(w_{j}x,w_{h}z{w}_l{w}_k^{*})=(w_{j}x,w_{h}z\left(w_l{w}_k\right))=(w_{j}x,0)=0$ for each $k\ne l$, also

$(w_{h}z{w}_l,w_{j}x{w}_k)=(z{w}_l,w_h^{*}{w}_{j}x{w}_k)=(z{w}_l,\left(w_h{w}_j\right)x{w}_k)=(z{w}_l,0)=0$ for each $j\ne h$.

Thus, the set $\{w_{j,k}:j,k\}$ is complete and $(w_{j,k}H,w_{h,l}H)=\left(0\right)$ for each $j\ne h$ or $k\ne l$, where the latter property is interpreted as the orthogonality. Together with property $\left(14\right)$, this implies that each element $x\in A$ has the form $x={\sum }_{j,k\in \mathsf{\Lambda }}{w}_{j,k}{x}_{j,k}$ with ${lim}_{j,k}{w}_{j,k}{x}_{j,k}=0$, since $A_H$ is the right H-module, also A is isomorphic with $A_G$ as the F-algebra and the right G-module, where the series may be infinite, $x_{j,k}\in H$ for each $j,k\in \mathsf{\Lambda }$, where $\mathsf{\Lambda }$ denotes the corresponding set.

Take the Banach H-bimodule $X_H=c_0(\mathsf{\Lambda },H)$ and to each element $x\in B$ one can pose the operator $T_x$ such that $e_j^{\prime }{T}_x{e}_k=x_{j,k}{\xi }_{j,k}$ (see Remark 3), where ${\xi }_{j,k}\in F$ and $|{\xi }_{j,k}|=|w_{j,k}|$ for each j and k in $\mathsf{\Lambda }$, since $\left|a\right|\in ({\mathsf{\Gamma }}_F\cup \left\{0\right\})$ for each $a\in A$, where $B:={\sum }_{j,k}{w}_{j}A{w}_k$ (see above). Then, $T_x\in L_{r,d}(X_H,X_H)$ and the mapping $T:B\to L_{r,d}(X_H,X_H)$ is the isometry having the isometrical extension $T:A\to L_{r,d}(X_H,X_H)$. The property $w_{j,k}{w}_{j,k}^{*}=w_j\ne 0$ given above provides $|w_{j,k}|\ne 0$ for each j and $k\in \mathsf{\Lambda }$, consequently, T is bijective from A onto $L_{r,d}(X_H,X_H)$, since A is simple.

For each S and V in $L_{r,d}(X_H,X_H)$, one has $SV\left(xb\right)=S\left(Vx\right)b=\left(SVx\right)b$ for each $b\in H$ and $x\in X_H$. Moreover, ${|\left(SV\right)}_{j,k}|\le {sup}_m|S_{j,m}\left|\right|V_{m,k}|$, consequently, $SV$ satisfies condition $\left(i\right)$ in Remark 3, that is $SV\in L_{r,d}(X_H,X_H)$. Hence, by verifying other properties, one gets that $L_{r,d}(X_H,X_H)$ also has the F-algebra structure. From the construction of $A_H$, it follows that $A_H$ is the F-algebra, since H and A are F-algebras. Notice that, moreover, $A_H$ as the F-algebra is isomorphic with the Banach F-algebra $L_{r,d}(X_H,X_H)$. By the conditions of this theorem, $A_H$ is isomorphic with A as the F-algebra and the right H-module. □

Theorem 4.

Let A be a spherically complete simple unital $B^{*}$-algebra over the spherically complete field F with $R_c\left(A\right)=R\left(A\right)$ and $Z\left(A\right)=F$. Let also G be a division algebra provided by Theorem 2 in [10] such that $s^{1/2}\in G$ for each $s\in G$. Then a division subalgebra H of G and a Banach H-bimodule $X_H$ exist such that $A_H$ and $L_{r,d}(X_H,X_H)$ are isomorphic as the Banach right H-modules and as F-algebras.

Proof. 

In this case, $H=G\cap G^{*}$ and instead of A we consider $A_H=A{\widehat{\otimes }}_{F}H$.

The $B^{*}$-algebra A is simple and central, $Z\left(A\right)=F$, hence the right H-module $A_H$ is simple due to Satz 5.9 in [7] and Theorem 2 above. We denote $A_H$ shortly by A and the rest of the proof is similar to that of Theorem 3. □

From Theorems 1, 3 and 4, the corollary follows.

Corollary 1.

Suppose that A is a spherically complete unital $B^{*}$-algebra over the spherically complete field F with $R_c\left(A\right)=R\left(A\right)$ and G is the division algebra given by Theorem 2 in [10] so that $s^{1/2}\in G$ for each $s\in G$ such that either

$\left(19\right)$$G\subset A$ and $G^{*}=G$ or

$\left(20\right)$$Z\left(A\right)=F$.

Then, a division subalgebra H in G with $H^{*}=H$ and H-bimodules $X_{k,H}$ exist such that $A_H$ as the right H-module and the F-algebra is the direct sum of $L_{r,d}(X_{k,H},X_{k,H})$.

Example 6.

Let A be a $B^{*}$-algebra over a spherically complete normed field F (see Definition 4 and Introduction). Evidently, the algebra A also has the structure of the Banach A-bimodule. Hence, there exists a Banach space H over F such that A can be embedded into the normed algebra $L(H,H)$ of all bounded F-linear operators $D:H\to H$. In view of Theorems 5.13 and 5.16 in [11], there exists a set α such that H is isomorphic with the Banach space $c_0(\alpha ,F)$ (see Remark 3). Therefore, each element D of A is characterized by the corresponding to it matrix $\left[D\right]$, which is unique relative to a fixed basis in H. This matrix is infinite, if $card\left(\alpha \right)\ge {\aleph }_0$.

5. Conclusions

The results obtained in this article can be used for further studies of infinite matrices structure. Moreover, it provides new tools for investigations of their algebras over normed fields, linear operator algebras on Banach spaces, spectral theory of linear operators, the representation theory of groups, algebraic geometry, PDEs, mathematical physics. Then, studies of relations with symplectic structures may be of some interest [13]. It is important also for their applications in the sciences, including quantum mechanics, quantum field theory, informatics, etc. (see [1,4,5,7,8,11,14,15,19,20,23,24,25] and references therein).