Algebra Seminar 

10/04/2002, 11:00 — 12:00 — Room P3.10, Mathematics Building
C. Martin Edwards, The QueenŽs College, Oxford, United Kingdom

The algebraic structure of bounded symmetric domains

The open unit ball in a complex Banach space $A$ is a bounded complex domain, the holomorphic structure of which leads to the existence of a closed subspace $A_s$ of $A$ and a triple product $\{...\}:A\times A_s\times A\to A$ which is symmetric and bilinear in the outer variables, conjugate linear in the middle variable and, for $a$, $b$ and $d$ in $A_s$ and $c$ in $A$, satisfies the Jordan triple identity,

${\displaystyle [D(a,b),D(c,d)]=D(\{abc\},d)-D(c,\{dab\}),}$

where $D(a,b)$ is the linear operator on $A$ defined, for $a$ in $A$ and $b$ in $A_s$ by

${\displaystyle D(a,b)e=\{abe\}.}$

When the bounded domain is symmetric $A_s$ exhausts $A$ and $A$ is then said to be a JB ${}^{*}$-triple. Hence, the study of JB ${}^{*}$-triples is equivalent to the study of bounded symmetric domains. A complex vector space satisfying the algebraic properties described above is said to be a Jordan ${}^{*}$-triple. An associative algebra $A$, with involution $a\to a^{*}$ and triple product

${\displaystyle \{abc\}=\frac{1}{2}({\mathrm{ab}}^{*}c+{\mathrm{cb}}^{*}a),}$

is an example of a Jordan ${}^{*}$-triple, as is the family of rectangular complex matrices with the same triple product. The algebraic structure of a Jordan ${}^{*}$-triple $A$ may be investigated by studying either its family $U(A)$ of tripotents or its family $I(A)$ of inner ideals, which are subspaces $J$ of $A$ for which the space $\{JAJ\}$ is contained in $J$. The family $I(A)$ contains the family $ZI(A)$ of triple ideals $J$ for which the spaces $\{AAJ\}$ and $\{AJA\}$ are contained in $J$. In some sense $ZI(A)$ is the centre of $I(A)$ and it is the consequences of this that form the main theme of the talk.