# Banach-Jordan algebra

Jordan–Banach algebra

A Jordan algebra over the field of real or complex numbers, endowed with a complete norm $|.|$ satisfying

\begin{equation*} \| x \circ y \| \leq \| x \| \| y \| \end{equation*}

for all $x$, $y$ in the algebra. Since an (associative) Banach algebra is a Banach–Jordan algebra under the Jordan product $x \circ y : = ( x y + y x ) / 2$, the theory of Banach–Jordan algebras can be regarded as a generalization of that of Banach algebras. For forerunners in this last theory, see Banach algebra and [a5]. Pioneering papers on Banach–Jordan algebras are [a4], [a19] and [a13]. A relatively complete panoramic view of the results on Banach–Jordan algebras can be obtained by combining [a16], [a3] and [a7].

Spectral methods in Banach–Jordan algebras have been possible thanks to the concept of invertible element in a Jordan algebra with a unit, introduced by N. Jacobson and K. McCrimmon (see [a12] or Jordan algebra). From this concept, the spectrum $\operatorname { sp } ( J , x )$ of an arbitrary element $x$ of a Banach–Jordan algebra $J$ is defined as in the associative case, and the spectral radius formula $\operatorname { lim } \{ \| x ^ { n } \| ^ { 1 / n } \} = \operatorname { max } \{ | \lambda | : \lambda \in \operatorname { sp } ( J , x ) \}$ holds. In fact, Banach–Jordan algebras are "locally spectrally" associative. This means that each element in such an algebra $J$ can be imbedded in some closed associative subalgebra $J ^ { \prime }$ of $J$ satisfying $\operatorname { sp } ( J , x ) = \operatorname { sp } ( J ^ { \prime } , x )$ for every $x \in J ^ { \prime }$. Then, for a single element in a complex Banach–Jordan algebra, a holomorphic functional calculus follows easily.

A Jordan algebra is said to be semi-simple (or semi-primitive, as preferred by people working in pure algebra) whenever its Jacobson-type radical [a11] is zero (cf. also Jacobson radical). Refining spectral methods, B. Aupetit [a2] gave a Jacobson-representation-theory-free proof of Johnson's uniqueness-of-norm theorem for semi-simple Banach algebras, and extended the result to semi-simple Banach–Jordan algebras. The absence of representation theory in Aupetit's proof was relevant because, although semi-simple Jordan algebras can be expressed as subdirect products of Jordan algebras which are "primitive" (in a peculiar Jordan sense), primitive Jordan algebras were not well-understood at that time. Aupetit's methods have shown also useful in extending from Banach algebras to Banach–Jordan algebras many other relevant results (see again [a3]), as well as in obtaining a general non-associative variant of Johnson's theorem [a15]. Recently, using work of E.I. Zel'manov [a22] on Jordan algebras without any finiteness condition, primitive Banach–Jordan algebras have been described in detail [a8]. Such a description has allowed one to extend to Banach–Jordan algebras the Johnson–Sinclair theorem, stating that derivations on semi-simple Banach algebras (cf. also Derivation in a ring) are automatically continuous [a18].

$\operatorname{JB}$-algebras are defined as the real Banach–Jordan algebras $J$ satisfying $\| x \| ^ { 2 } \leq \| x ^ { 2 } + y ^ { 2 } \|$ for all $x , y \in J$. The basic theory of $\operatorname{JB}$-algebras, originally due to E.M. Alfsen, F.W. Shultz and E. Stormer [a1], is fully treated in [a10]. If $A$ is a $C ^ { * }$-algebra, then the self-adjoint part $A _ { \text{sa} }$ of $A$ is a $\operatorname{JB}$-algebra under the Jordan product. Closed subalgebras of $A _ { \text{sa} }$, for some $C ^ { * }$-algebra $A$, become relevant examples of $\operatorname{JB}$-algebras, and are called $\operatorname {JC}$-algebras. Through the consideration of $\text{JBW}$-algebras (i.e., $\operatorname{JB}$-algebras that are dual Banach spaces, cf. also Banach space), $\text{JBW}$-factors (i.e., prime $\text{JBW}$-algebras), and factor representations of a given $\operatorname{JB}$-algebra $J$ (i.e., $w ^ { * }$-dense range homomorphisms from $J$ to $\text{JBW}$-factors), the knowledge of arbitrary $\operatorname{JB}$-algebras is reasonably reduced to that of $\operatorname {JC}$-algebras and the exceptional $\operatorname{JB}$-algebra $H _ { 3 } ( \text{O} )$ of all Hermitian $( 3 \times 3 )$-matrices over the alternative division algebra $\mathbf{O}$ of real octonions.

$\operatorname{JB} ^ { * }$-algebras are defined as complex Banach–Jordan algebras $J$ endowed with a conjugate-linear algebra involution $*$ satisfying $\| U _ { X } ( x ^ { * } ) \| = \| x \| ^ { 3 }$ for every $x \in J$. Here, for $x \in J$, $U _ { x }$ denotes the operator on $J$ defined by $U _ { x } ( y ) := 2 x \circ ( x \circ y ) - x ^ { 2 } \circ y$ for every $y \in J$. Every $C ^ { * }$-algebra becomes a $\operatorname{JB} ^ { * }$-algebra under its Jordan product. $\operatorname{JB} ^ { * }$-algebras are closely related to $\operatorname{JB}$-algebras. Indeed, $\operatorname{JB}$-algebras are nothing but the self-adjoint parts of $\operatorname{JB} ^ { * }$-algebras [a20]. The one-to-one categorical correspondence between $\operatorname{JB}$-algebras and $\operatorname{JB} ^ { * }$-algebras derived from the above result completely reduces the $*$-theory of $\operatorname{JB} ^ { * }$-algebras to the theory of $\operatorname{JB}$-algebras. However, $\operatorname{JB} ^ { * }$-algebras are of interest on their own, mainly due to their connection with complex analysis (see [a6], [a17], and [a21]). Using Zel'manov's prime theorem, the structure theory of $\operatorname{JB}$- and $\operatorname{JB} ^ { * }$-algebras can be refined as follows (see [a9]). A $\operatorname{JB} ^ { * }$-algebra $J$ is primitive if and only if it is of one of the following types:

$J$ is the unique $\operatorname{JB} ^ { * }$-algebra whose self-adjoint part is $H _ { 3 } ( \text{O} )$.

There exists a complex Hilbert space $( H , ( \cdot | \cdot ) )$ of dimension $\geq 3$, with a conjugation $\sigma$ and a $\sigma$-invariant norm-one element $1$, such that $J = H$ as complex vector spaces, whereas the product $\circ$, the involution $*$, and the norm $|.|$ of $J$ are given by

\begin{equation*} x \circ y : = ( x | 1 ) y + ( y | 1 ) x - ( x | \sigma ( y ) ) 1, \end{equation*}

\begin{equation*} x ^ { * } : = 2 ( 1 | x ) 1 - \sigma ( x ) , \| x \| ^ { 2 } : = ( x | x ) + ( ( x | x ) ^ { 2 } - | ( x | \sigma ( x ) ) | ^ { 2 } ) ^ { 1 / 2 }, \end{equation*}

respectively.

There exists a primitive $C ^ { * }$-algebra $A$ such that $J$ is a closed self-adjoint Jordan subalgebra of the $C ^ { * }$-algebra $M ( A )$, of multipliers of $A$, containing $A$.

There exists a primitive $C ^ { * }$-algebra with a $*$-involution $\tau$ such that $J$ is a closed self-adjoint Jordan subalgebra of $M ( A )$ contained in the $\tau$-Hermitian part of $M ( A )$ and containing the $\tau$-Hermitian part of $A$.

From the point of view of analysis, the Jordan identity $x \circ ( y \circ x ^ { 2 } ) = ( x \circ y ) \circ x ^ { 2 }$ (which, together with the commutativity, is characteristic of Jordan algebras) can be regarded as a theorem instead of as an axiom. Indeed, if a unital complete normed non-associative complex algebra $A$ is subjected to the geometric conditions that, through the Vidav–Palmer theorem, characterize $C ^ { * }$-algebras in the associative setting, then $A$ under the product $x \circ y : = ( x y + y x ) / 2$ and a suitable involution becomes a $\operatorname{JB} ^ { * }$-algebra [a14].

 [a1] E.M. Alfsen, F.W. Shultz, E. Stormer, "A Gelfand–Neumark theorem for Jordan algebras" Adv. Math. , 28 (1978) pp. 11–56 [a2] B. Aupetit, "The uniqueness of the complete norm topology in Banach algebras and Banach Jordan algebras" J. Funct. Anal. , 47 (1982) pp. 7–25 [a3] B. Aupetit, "Recent trends in the field of Jordan–Banach algebras" J. Zemánek (ed.) , Functional Analysis and Operator Theory , 30 , Banach Center Publ. (1994) pp. 9–19 [a4] V.K. Balachandran, P.S. Rema, "Uniqueness of the complete norm topology in certain Banach Jordan algebras" Publ. Ramanujan Inst. , 1 (1969) pp. 283–289 [a5] F.F. Bonsall, J. Duncan, "Complete normed algebras" , Springer (1973) [a6] R.B. Braun, W. Kaup, H. Upmeier, "A holomorphic characterization of Jordan $C ^ { * }$-algebras" Math. Z. , 161 (1978) pp. 277–290 [a7] M. Cabrera, A. Moreno, A. Rodriguez, "Normed versions of the Zel'manov prime theorem: positive results and limits" A. Gheondea (ed.) R.N. Gologan (ed.) D. Timotin (ed.) , Operator Theory, Operator Algebras and Related Topics (16th Internat. Conf. Operator Theory, Timisoara (Romania) July, 2-10, 1996) , The Theta Foundation, Bucharest (1997) pp. 65–77 [a8] M. Cabrera, A. Moreno, A. Rodriguez, "Zel'manov's theorem for primitive Jordan–Banach algebras" J. London Math. Soc. , 57 (1998) pp. 231–244 [a9] A. Fernandez, E. Garcia, A. Rodriguez, "A Zelmanov prime theorem for $J B ^ { * }$-algebras" J. London Math. Soc. , 46 (1992) pp. 319–335 [a10] H. Hanche-Olsen, E. Stormer, "Jordan operator algebras" , Monograph Stud. Math. , 21 , Pitman (1984) [a11] L. Hogben, K. Mccrimmon, "Maximal modular inner ideals and the Jacobson radical of a Jordan algebra" J. Algebra , 68 (1981) pp. 155–169 [a12] N. Jacobson, "Structure and representations of Jordan algebras" , Colloq. Publ. , 37 , Amer. Math. Soc. (1968) [a13] P.S. Putter, B. Yood, "Banach Jordan $*$-algebras" Proc. London Math. Soc. , 41 (1980) pp. 21–44 [a14] A. Rodriguez, "Nonassociative normed algebras spanned by hermitian elements" Proc. London Math. Soc. , 47 (1983) pp. 258–274 [a15] A. Rodriguez, "The uniqueness of the complete algebra norm topology in complete normed nonassociative algebras" J. Funct. Anal. , 60 (1985) pp. 1–15 [a16] A. Rodriguez, "Jordan structures in analysis" W. Kaup (ed.) K. McCrimmon (ed.) H.P. Petersson (ed.) , Jordan Algebras (Proc. Conf. Oberwolfach, Germany, August, 9-15, 1992) , de Gruyter (1994) pp. 97–186 [a17] H. Upmeier, "Symmetric Banach manifolds and Jordan $C ^ { * }$-algebras" , North-Holland (1985) [a18] A.R. Villena, "Continuity of derivations on Jordan–Banach algebras." Studia Math. , 118 (1996) pp. 205–229 [a19] C. Viola Devapakkiam, "Jordan algebras with continuous inverse" Math. Japon. , 16 (1971) pp. 115–125 [a20] J.D. M. Wright, "Jordan $C ^ { * }$-algebras" Michigan Math. J. , 24 (1977) pp. 291–302 [a21] M.A. Youngson, "Non unital Banach Jordan algebras and $C ^ { * }$-triple systems" Proc. Edinburgh Math. Soc. , 24 (1981) pp. 19–31 [a22] E. Zel'manov, "On prime Jordan algebras II" Sib. Math. J. , 24 (1983) pp. 89–104