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In [[mathematics]], '''Jordan operator algebras''' are real or complex [[Jordan algebra]]s with the compatible structure of a Banach space. When the coefficients are [[real numbers]], the algebras are called '''Jordan Banach algebras'''. The theory has been extensively developed only for the subclass of '''JB algebras'''. The axioms for these algebras were devised by {{harvtxt|Alfsen|Schultz|Størmer|1978}}. Those that can be realised concretely as subalgebras of self-adjoint operators on a real or complex Hilbert space with the operator Jordan product and the [[operator norm]] are called '''JC algebras'''. The axioms for complex Jordan operator algebras, first suggested by [[Irving Kaplansky]] in 1976, require an involution and are called '''JB* algebras''' or '''Jordan C* algebras'''. By analogy with the abstract characterisation of [[von Neumann algebra]]s as [[C* algebra]]s for which the underlying Banach space is the dual of another, there is a corresponding definition of '''JBW algebras'''. Those that can be realised using [[weak operator topology|ultraweakly closed]] Jordan algebras of self-adjoint operators with the operator Jordan product are called '''JW algebras'''. The JBW algebras with trivial center, so-called '''JBW factors''', are classified in terms of von Neumann factors: apart from the exceptional 27 dimensional [[Albert algebra]] and the '''spin factors''', all other JBW factors are isomorphic either to the self-adjoint part of a von Neumann factor or to its fixed point algebra under a period two *-antiautomorphism. Jordan operator algebras have been applied in [[quantum mechanics]] and in [[complex geometry]], where [[Max Koecher|Koecher's]] description of [[bounded symmetric domain]]s using [[Jordan algebra]]s has been extended to infinite dimensions.
==Definitions==
===JC algebra===
A '''JC algebra''' is a real subspace of the space of self-adjoint operators on a real or complex Hilbert space, closed under the operator Jordan product ''a'' ∘ ''b'' = ½(''ab'' + ''ba'') and closed in the operator norm.

===Jordan Banach algebra===
A '''Jordan Banach algebra''' is a real Jordan algebra with a norm making it a Banach space and satisfying || ''a'' ∘ ''b'' || ≤ ||''a''||⋅||''b''||.
===JB algebra===
A '''JB algebra''' is a Jordan Banach algebra satisfying

:<math>\displaystyle{\|a^2\|=\|a\|^2,\,\,\, \|a^2\| \le \|a^2 + b^2\|.}</math>
===JB* algebras===
A '''JB* algebra''' or '''Jordan C* algebra''' is a complex Jordan algebra with an involution ''a'' ↦ ''a''* and a norm making it a Banach space and satisfying

* ||''a'' ∘ ''b'' || ≤ ||''a''||⋅||''b''||
* ||''a''*|| = ||''a''||
* ||{''a'',''a''*,''a''}|| = ||''a''||3 where the [[Jordan triple system|Jordan triple product]] is defined by {''a'',''b'',''c''} = (''a'' ∘ ''b'') ∘ ''c'' + (''c'' ∘ ''b'') ∘ ''a'' − (''a'' ∘ ''c'') ∘ ''b''.

===JW algebras===
A '''JW algebra''' is a Jordan subalgebra of the Jordan algebra of self-adjoint operators on a complex Hilbert space that is closed in the [[weak operator topology]].

===JBW algebras===
A '''JBW algebra''' is a JB algebra that, as a real Banach space, is the dual of a Banach space called its '''predual'''.<ref>{{harvnb|Hanche-Olsen|Størmer|1984|p=111}}</ref> There is an equivalent more technical definition in terms of the continuity properties of the linear functionals in the predual, called '''normal functionals'''. This is usually taken as the definition and the abstract characterization as a dual Banach space derived as a consequence.<ref>{{harvnb|Hanche-Olsen|Størmer|1984|p=94}}</ref>

* For the order structure on a JB algebra (defined below), any increasing net of operators bounded in norm should have a least upper bound.
* Normal functionals are those that are continuous on increasing bounded nets of operators. Positive normal functional are those that are non-negative on positive operators.
* For every non-zero operator, there is a positive normal functional that does not vanish on that operator.
==Properties of JB algebras==
{{Seealso|Euclidean Jordan algebra}}
*If a unital JB algebra is [[associative algebra|associative]], then its complexification with its natural involution is a commutative C* algebra. It is therefore isomorphic to C(''X'') for a compact Hausdorff space ''X'', the space of characters of the algebra.
*'''Spectral theorem.''' If ''a'' is a single operator in a JB algebra, the closed subalgebra generated by 1 and ''a'' is associative. It can be identified with the continuous real-valued functions on the spectrum of ''a'', the set of real λ for which ''a'' − λ1 is not invertible.
*The positive elements in a unital JB algebra are those with spectrum contained in [0,∞). By the spectral theorem, they coincide with the space of squares and form a closed convex cone. If ''b'' ≥ 0, then {''a'',''b'',''a''} ≥ 0.
* A JB algebra is a [[formally real Jordan algebra]]: if a sum of squares of terms is zero, then each term is zero. In finite dimensions, a JB algebra is isomorphic to a [[Euclidean Jordan algebra]].<ref>{{harvnb|Faraut|Koranyi|1994}}</ref>
* The [[spectral radius]] on a JB algebra defines an equivalent norm also satisfying the axioms for a JB algebra.
* A state on a unital JB algebra is a bounded linear functional ''f'' such that ''f''(1) = 1 and ''f'' is non-negative on the positive cone. The state space is a convex set closed in the weak* topology. The extreme points are called pure states. Given ''a'' there is a pure state ''f'' such that |''f''(''a'')| = ||''a''||.
* '''[[Gelfand–Naimark–Segal construction]]''': If a JB algebra is isomorphic to the self-adjoint ''n'' by ''n'' matrices with coefficients in some associative unital *-algebra, then it is isometrically isomorphic to a JC algebra. The JC algebra satisfies the additional condition that (''T'' + ''T''*)/2 lies in the algebra whenever ''T'' is a product of operators from the algebra.<ref>{{harvnb|Hanche-Olsen|Størmer|1984|pp=75–90}}</ref>
* A JB algebra is '''purely exceptional''' if it has no non-zero Jordan homomorphism onto a JC algebra. The only simple algebra that can arise as the homomorphic image of a purely exceptional JB algebra is the [[Hurwitz's theorem (composition algebras)#Applications to Jordan algebras|Albert algebra]], the 3 by 3 self-adjoint matrices over the [[octonion]]s.
*Every JB algebra has a uniquely determined closed ideal that is purely exceptional, and such that the quotient by the ideal is a JC algebra.
*'''Shirshov–Cohn theorem.''' A JB algebra generated by 2 elements is a JC algebra. <ref>{{harvnb|Hanche-Olsen|Størmer|1984|pp=155–156}}</ref>

==Properties of JB* algebras==
The definition of JB* algebras was suggested in 1976 by [[Irving Kaplansky]] at a lecture in Edinburgh. The real part of a JB* algebra is always a JB algebra. {{harvtxt|Wright|1977}} proved that conversely the complexification of every JB algebra is a JB* algebra. JB* algebras have been used extensively as a framework for studying bounded symmetric domains in infinite dimensions. This generalizes the theory in finite dimensions developed by [[Max Koecher]] using the [[Symmetric cone#Complexification of Euclidean Jordan algebra|complexification of a Euclidean Jordan algebra]]. <ref>See:
*{{harvnb|Hanche-Olsen|Støormer|1984|p=90-92}}
*{{harvnb|Upmeier|1985}}</ref>

==Properties of JBW algebras==
===Elementary properties===
* The [[Kaplansky density theorem]] holds for real unital Jordan algebras of self-adjoint operators on a Hilbert space with the operator Jordan product. In particular a Jordan algebra is closed in the [[weak operator topology]] if and only if it is closed in the [[Ultraweak topology|ultraweak operator topology]]. The two topologies coincide on the Jordan algebra. <ref>See:
*{{harvnb|Effros|Størmer|1967}}
*{{harvnb|Dixmier|1957}}
*{{harvnb|Dixmier|1981}}
*{{harvnb|Hanche-Olsen|Størmer|1984|p=112}}</ref>
*For a JBW algebra, the space of positive normal functionals is invariant under the quadratic representation ''Q''(''a'')''b'' = {''a'',''b'',''a''}. If ''f'' is positive so is ''f'' ∘ ''Q''(''a'').
*The weak topology on a JW algebra ''M'' is define by the seminorms |''f''(''a'')| where ''f'' is a normal state; the strong topology is defined by the seminorms |''f''(''a''2)|1/2. The quadratic representation and Jordan product operators ''L''(''a'')''b'' = ''a'' ∘ ''b'' are continuous operators on ''M'' for the both the weak and strong topology.
*An idempotent ''p'' in a JBW algebra ''M'' is called a '''projection'''. If ''p'' is a projection, then ''Q''(''p'')''M'' is a JBW algebra with identity ''p''.
*If ''a'' is any element of a JBW algebra, the smallest weakly closed unital subalgebra it generates is associative and hence the self-adjoint part of an Abelian von Neumann algebra. In particular ''a'' can be approximated in norm by linear combinations of orthogonal projections.
*The projections in a JBW algebra are closed under lattice operations. Thus for an family ''p''α there is a smallest projection ''p'' such that ''p'' ≥ ''p''sub>α and a largest projection ''q'' such that ''q'' ≤ 'p''sub>α.
*The '''center''' of a JBW algebra ''M'' consists of all ''z'' such ''L''(''z'') commutes with ''L''(''a'') for ''a'' in ''M''. It is an associative algebra and the real part of an Abelian von Neumann algebra. A JBW algebra is called a '''factor''' if its center consists of scalar operators.
*If ''A'' is a JB algebra, its second dual ''A''** is a JBW algebra. The normal states are states in ''A''* and can be identified with states on ''A''. Moreover ''A''** is the JBW algebra generated by ''A''.
*A JB algebra is a JBW algebra if and only if, as a real Banach space, it is the dual of a Banach space. This Banach space, its ''predual'', is the space of normal functionals, defined as differences of positive normal functionals. These are the functionals continuous for the weak or strong topologies. As a consequence the weak and strong topologies coincide on a JBW algebra.
*In a JBW algebra, the JBW algebra generated by a Jordan subalgebra coincides with its weak closure. Moreover an extension of the Kaplansky density theorem holds: the unit ball of the subalgebra is weakly dense in the unit ball of the JBW algebra it generates.
*[[Tomita–Takesaki theory]] has been extended by {{harvtxt|Haagerup|Hanche-Olsen|1984}} to normal states of a JBW algebra that are faithful, i.e. do not vanish on any non-zero positive operator. The theory can be deduced from the original theory for von Neumann algebras.<ref>{{harvnb|Hanche-Olsen|Størmer|1984|pp=94–119}}</ref>

===Comparison of projections===
Let ''M'' be a JBW factor. The inner automorphisms of ''M'' are those generated by the period two automorphisms ''Q''(1 – 2''p'') where ''p'' is a projection. Two projections are equivalent if there is an inner automorphism carrying one onto the other. Given two projections in a factor, one of them is always equivalent to a subprojection of the other. If each is equivalent to a subprojection of the other, they are equivalent.

A JBW factor can be classified into three mutually exclusive types as follows:

*It is type I if there is a minimal projection. It is type I''n'' if 1 can be written as a sum of ''n'' orthognal minimal projections for 1 ≤ ''n'' ≤ ∞.
*It is Type II if there are no minimal projections but the subprojections of some fixed projections ''e'' form a [[modular lattice]], i.e. ''p'' ≤ ''q'' implies (''p'' ∨ ''r'') ∧ ''q'' = ''p'' ∨ (''r'' ∧ ''q'') for any projection ''r'' ≤ ''e''. If ''e'' can be taken to be 1, it is Type II1. Otherwise it is type II≈.
*It is Type III if the projections do not form a modular lattice. All non-zero projections are then equivalent.<ref>{{harvnb|Hanche-Olsen|Størmer|120–134}}</ref>

Tomita−Takesaki theory permits a further classification of the type III case into types IIIλ (0 ≤ λ ≤ 1) with the additional invariant of an [[ergodic flow]] on a [[Lebesgue space]] (the "flow of weights") when λ = 0.<ref>{{harvnb|Haagerup|Hanche-Olsen|1984}}</ref>

===Classification of JBW factors of Type I===
* The JBW factor of Type I1 is the [[real numbers]].
* The JBW factors of Type I2 are the '''spin factors'''. Let ''H'' be a real Hilbert space of dimension greater than 1. Set ''M'' = ''H'' ⊕ '''R''' with inner product (''u''⊕λ,''v''⊕μ) =(''u'',''v'') + λμ and product (u⊕λ)∘(v⊕μ)=( μ''u'' + λ''v'') ⊕ [(''u'',''v'') + λμ]. With the operator norm ||''L''(''a'')||, ''M'' is a JBW factor and also a JW factor.
* The JBW factors of Type I3 are the self-adjoint 3 by 3 matrices with entries in the real numbers, the [[complex number]]s or the [[quaternion]]s or the [[octonion]]s.
*The JBW factors of Type I''n'' with 4 ≤ ''n'' < ∞ are the self-adjoint ''n'' by ''n'' matrices with entries in the real numbers, the complex numbers or the quaternions.
*The JBW factors of Type I∞ are the self-adjoint operators on the unique infinire-dimensional real, complex or quaternionic Hilbert space. The quaternionic space is defined as all sequences '''x''' = (''x''''i'') with ''x''''i'' in '''H''' and ∑ |''x''''i''|2 < ∞. The '''H'''-valued inner product is given by ('''x''','''y''') = ∑ (''y''''i'')*''x''''i''. There is an underlying real inner product given by ('''x''','''y''')'''R''' = Re ('''x''','''y'''). The quaternionic JBW factor of Type I∞ is thus the Jordan algebra of all self-adjoint operators on this real inner product space that commute with the action of right multiplication by '''H'''.<ref>{{harvnb|Hanche-Olsen|Størmer|1984}}</ref>

===Classification of JBW factors of Types II and III===
The JBW factors not of Type I2 and I3 are all JW factors, i.e. can be realized as Jordan algebras of self-adjoint operators on a Hilbert space closed in the weak operator topology. Every JBW factor not of Type I2 or Type I3 is isomorphic to the self-adjoint part of the fixed point algebra of a period 2 *-antiautomorphism of a von Neumann algebra. In particular each JBW factor is either isomorphic to the self-adjoint part of a von Neumann factor of the same type or to the self-adjoint part of the fixed point algebra of a period 2 *-antiautomorphism of a von Neumann factor of the same type. <ref>See:
*{{harvnb|Hanche-Olsen|Størmer|1984|pp=122–123}}
*{{harvnb|Hanche-Olsen|1983}}
*{{harvnb|Haagerup|Hanche-Olsen|1984|p=347}}
</ref> For [[von Neumann algebra#Amenable von Neumann algebras|hyperfinite factors]], the class of von Neumann factors completely classified by [[Alain Connes|Connes]] and Haagerup, the period 2 *-antiautomorphisms have been classified up to conjugacy in the automorphism group of the factor.<ref> See:
*{{harvnb|Størmer|1980}}
*{{harvnb|Giordano|Jones|1980}}
*{{harvnb|Giordano|1983a}}
*{{harvnb|Giordano|1983b}}</ref>

==See also==
*[[Jordan algebra]]
*[[Euclidean Jordan algebra]]

==Notes==
{{reflist|2}}

==References==
*{{citation|last=Alfsen|first= E. M.|last2= Shultz|first2= F. W.|last3= Størmer|first3= E.|title=A Gelfand-Neumark theorem for Jordan algebras|journal=Advances in Math.|volume= 28|year=1978|pages= 11–56|url=http://www.sciencedirect.com/science/article/pii/0001870878900440}}
*{{citation|first=J.|last= Dixmier|title=Von Neumann algebras| isbn=0-444-86308-7 |year=1981}} (A translation of {{citation|first=J.|last= Dixmier|title=Les algèbres d'opérateurs dans l'espace hilbertien: algèbres de von Neumann|publisher= Gauthier-Villars |year=1957}}, the first book about von Neumann algebras.)
*{{citation|last=Effros|first= E. G.|last2= Størmer|first2= E.|title=Jordan algebras of self-adjoint operators|
journal=Trans. Amer. Math. Soc.|volume= 127|year= 1967|pages= 313–316}}
*{{citation|last=Faraut|first= J.|last2= Koranyi|first2= A.|title= Analysis on symmetric cones|series= Oxford Mathematical Monographs|publisher= Oxford University Press|year= 1994|isbn= 0198534779}} \
*{{citation|last=Giordano|first= T.|last2= Jones|first2= V.|title=Antiautomorphismes involutifs du facteur hyperfini de type II1|journal=C. R. Acad. Sci. Paris|year=1980|pages= A29–A31}}
*{{citation|last=Giordano|first= T.|title=
Antiautomorphismes involutifs des facteurs de von Neumann injectifs. I|journal=J. Operator Theory|volume= 10|year=1983a|pages= 251–287}}
*{{citation|last=Giordano|first= T.|title=
Antiautomorphismes involutifs des facteurs de von Neumann injectifs. II|journal= J. Funct. Anal.|volume= 51|year=1983b|pages= 326–360}}
*{{citation|last=Hanche-Olsen|first= H.|title=On the structure and tensor products of JC-algebras|journal=
Canad. J. Math.|volume= 35|year=1983|pages= 1059–1074|url=http://cms.math.ca/10.4153/CJM-1983-059-8}}
*{{citation|last=Haagerup|first= U.|last2= Hanche-Olsen|first2= H.|title= Tomita–Takesaki theory for Jordan algebras|journal= J. Operator Theory |volume=11|year=1984|pages= 343–364}}
*{{citation|last=Hanche-Olsen|first= H.|last2= Størmer|first2= E.|title=Jordan operator algebras|series=
Monographs and Studies in Mathematics|volume= 21|publisher=Pitman|year= 1984|id= ISBN 0273086197|url=http://www.math.ntnu.no/~hanche/joa/}}
*{{citation|last=Størmer|first= E.|title=Real structure in the hyperfinite factor| journal=Duke Math. J.|volume= 47|year=1980|pages=145–153}}
*{{citation|last=Upmeier|first=H.|title= Symmetric Banach manifolds and Jordan C∗-algebras|series=North-Holland Mathematics Studies|volume= 104|year= 1985|id=ISBN 0444876510}}
*{{citation|last=Upmeier|first= H.|title=Jordan algebras in analysis, operator theory, and quantum mechanics|series= CBMS Regional Conference Series in Mathematics|volume= 67|publisher=American Mathematical Society|year= 1987|id= ISBN 082180717X}}
{{citation|last=Wright|first= J. D. M.|title=Jordan C∗-algebras|journal=
Michigan Math. J.|volume= 24|year=1977|pages= 291–302}}

[[Category:Operator algebras]]
[[Category:Operator theory]]
[[Category:Functional analysis]]
[[Category:Non-associative algebras]]

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